Open AccessArticle

Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains

Muhammad Sabil Ur Rehman

^1,†,

Qazi Zahoor Ahmad

^1,†

Isra Al-shbeil

^2,†

Sarfraz Ahmad

^1,†,

Ajmal Khan

^1,†,

Bilal Khan

^3,†

and

Jianhua Gong

^4,*,†

Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan

Department of Mathematics, The University of Jordan, Amman 11942, Jordan

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

⁴

Department of Mathematical Sciences, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2022, 11(10), 494; https://doi.org/10.3390/axioms11100494

Submission received: 8 August 2022 / Revised: 10 September 2022 / Accepted: 15 September 2022 / Published: 23 September 2022

(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)

Download Versions Notes

Abstract

In the current work, by using the familiar q-calculus, first, we study certain generalized conic-type regions. We then introduce and study a subclass of the multivalent q-starlike functions that map the open unit disk into the generalized conic domain. Next, we study potentially effective outcomes such as sufficient restrictions and the Fekete–Szegö type inequalities. We attain lower bounds for the ratio of a good few functions related to this lately established class and sequences of the partial sums. Furthermore, we acquire a number of attributes of the corresponding class of q-starlike functions having negative Taylor–Maclaurin coefficients, including distortion theorems. Moreover, various important corollaries are carried out. The new explorations appear to be in line with a good few prior commissions and the current area of our recent investigation.

Keywords:

analytic functions; multivalent (p-valent) functions; q-derivative (q-difference) operator; differential subordination

MSC:

Primary 30C45; 30C50; 30C80; Secondary 11B65; 47B38

1. Introduction and Preliminaries

Let

A (p)

denote the multivalent (p-valent) class of functions j in the open unit disk

U

on the complex plane

C,

U = \{ζ : ζ \in C and |ζ| < 1\},

with the series expansion

j (ζ) = ζ^{p} + \sum_{n = 1}^{\infty} a_{n + p} ζ^{n + p} (p \in N = {1, 2, . . .}) .

(1)

It is clear that

A (1) = A,

where

A

is the class of normalized analytic functions j with

j (0) = 0 and j^{'} (0) = 1 .

(2)

The multivalent

(p - valent)

subclass of functions is signified by

S^{*} (p),

which comprises functions

j \in A (p)

satisfying

ℜ (\frac{ζ j^{'} (ζ)}{j (ζ)}) > 0 (ζ \in U) .

It can be observed that

S^{*} (1) = S^{*},

where

S^{*}

is the prominent class of starlike functions.

For two analytic functions j and g in

U

, it is affirmed in [1] that j is subordinate to g, expressed as

j ≺ g, or j (ζ) ≺ g (ζ),

if there exists a Schwarz function w, which is analytic in

U

with

w (0) = 0 and |w (ζ)| < 1

such that

j (ζ) = g (w (ζ)) .

Moreover, if j is subordinate to the analytic univalent function g in

U,

then

j (0) = g (0) and j (U) \subset g (U) .

Let

P

present the familiar Carathéodory class of analytic functions

Φ

U

, normalized by (see, for example, [2])

Φ (ζ) = 1 + \sum_{n = 1}^{\infty} c_{n} ζ^{n},

(3)

such that

ℜ \{Φ (ζ)\} > 0 (ζ \in U) .

Definition 1.

A function h with

h (0) = 1

is said to be in class

P [N, L]

h (ζ) ≺ \frac{1 + N ζ}{1 + L ζ} (- 1 ≦ L < N ≦ 1) .

In particular, the class

P [N, L]

of analytic functions was launched by Janowski [3], who established that

h (ζ) \in P [N, L]

if and only if there exists a function

Φ \in P

so that

h (ζ) = \frac{(N + 1) Φ (ζ) - (N - 1)}{(L + 1) Φ (ζ) - (L - 1)} (- 1 ≦ L < N ≦ 1) .

Definition 2.

A function

j \in A

is said to be in class

S^{*} [N, L]

\frac{ζ j^{'} (ζ)}{j (ζ)} = \frac{(N + 1) Φ (ζ) - (N - 1)}{(L + 1) Φ (ζ) - (L - 1)} (- 1 ≦ L < N ≦ 1) .

(4)

Kanas et al. [4,5,6] were the first who illustrated the conic domain

Ω_{k}

(k ≧ 0)

as follows

Ω_{k} = \{ζ = u + i v \in C : u > k \sqrt{{(u - 1)}^{2} + v^{2}}\}

(5)

and, subjected to this domain, they also initiated and examined the corresponding class k-

ST

of k-starlike functions (see Definition 3 below). In particular,

(a) If

k = 0,

Ω_{0}

acts on the conic region bounded sequentially by the imaginary axis;

(b) If

k = 1

Ω_{1}

is a parabola;

0 < k < 1

Ω_{k}

is the right-hand branch of hyperbola;

(d) If

k > 1

Ω_{k}

represents an ellipse.

For these conic regions, the following functions act as extremal functions:

φ_{k} (ζ) = \{\begin{matrix} \frac{1 + ζ}{1 - ζ} = 1 + 2 ζ + 2 ζ^{2} + \dots & (k = 0) \\ 1 + \frac{2}{π^{2}} {(log \frac{1 + \sqrt{ζ}}{1 - \sqrt{ζ}})}^{2} & (k = 1) \\ 1 + \frac{2}{1 - k^{2}} {sinh}^{2} \{(\frac{2}{π} arccos k) arctanh (\sqrt{ζ})\} & (0 < k < 1) \\ 1 + \frac{1}{k^{2} - 1} sin (\frac{π}{2 K (κ)} \int_{0}^{\frac{u (ζ)}{\sqrt{κ}}} \frac{d t}{\sqrt{1 - t^{2}} \sqrt{1 - κ^{2} t^{2}}}) + \frac{1}{k^{2} - 1} & (k > 1) \end{matrix}

(6)

where

u (ζ) = \frac{ζ - \sqrt{κ}}{1 - \sqrt{κ} ζ} (ζ \in U)

and

κ \in (0, 1)

is selected so that

k = cosh (π K^{'} (κ) / (4 K (κ)))

. Here,

K (κ)

is Legendre’s complete elliptic integral of the first type and

K^{'} (κ) = K (\sqrt{1 - κ^{2}}

), i.e.,

K^{'} (κ)

is the complementary integral of

K (κ)

It was proven in [7] that if we assume that

φ_{k} (ζ) = 1 + P_{1} ζ + P_{2} ζ^{2} + \dots (ζ \in U),

then we have

P_{1} = \{\begin{matrix} \frac{2 N^{2}}{1 - k^{2}} & (0 ≦ k < 1) \\ \frac{8}{π^{2}}, & (k = 1) \\ \frac{π^{2}}{4 k^{2} {(κ)}^{2} (1 + κ) \sqrt{κ}} & (k > 1) \end{matrix}

(7)

and

P_{2} = D (k) P_{1},

(8)

where

D (k) = \{\begin{matrix} \frac{N^{2} + 2}{3} & (0 ≦ k < 1) \\ \frac{8}{π^{2}} & (k = 1) \\ \frac{{[4 K (κ)]}^{2} (κ^{2} + 6 κ + 1) - π^{2}}{24 {[K (κ)]}^{2} (1 + κ) \sqrt{κ}} & (k > 1) \end{matrix}

(9)

with

N = \frac{2}{π} arccos k .

These conic regions were formulated and generalized by many authors; for instance see [8,9,10].

The class k-

ST

is defined as below.

Definition 3.

A function

j \in

A

is said to be considered to be in the class

k - ST

, if

\frac{ζ j^{'} (ζ)}{j (ζ)} ≺ φ_{k} (ζ) (ζ \in U; k ≧ 0),

where

φ_{k} (ζ)

is given by (6).

Noor et al. [11] amalgamated the idea of the Janowski functions and the conic regions, and introduced the following definition.

Definition 4.

A function

h \in P

is said to be in class k-

P [N, L]

h (ζ) ≺ \frac{(N + 1) φ_{k} (ζ) - (N - 1)}{(L + 1)) φ_{k} (ζ) - (L - 1)} (- 1 ≦ L < N ≦ 1; k ≧ 0)

(10)

where

φ_{k} (ζ)

is defined by (6).

Geometrically, each function

h \in k

P [N, L]

takes all points in the domain

Ω_{k} [N, L]

(- 1 ≦ L < N ≦ 1; k ≧ 0)

, which is defined as follows

Ω_{k} [N, L] = \{w \in C : ℜ (\frac{(L - 1) w - (N - 1)}{(L + 1)) w - (N + 1)}) > k |\frac{(L - 1) w - (N - 1)}{(L + 1)) w - (N + 1)} - 1|\} .

Thus,

Ω_{k} [N, L]

is a set of complex numbers

w = u + i v

such that

\begin{matrix} {[(L^{2} - 1) (u^{2} + v^{2}) - 2 (N L - 1) u + (N^{2} - 1)]}^{2} \\ > k [{(- 2 (L + 1)) (u^{2} + v^{2}) + 2 (N + L + 2) u - 2 (N + 1))}^{2} + 4 {(N - L)}^{2} v^{2}] . \end{matrix}

Domain

Ω_{k} [N, L]

depicts the conic-type regions in detail (see [11]).

Definition 5.

A function

j \in A

is said to be in class k-

ST [N, L]

\frac{ζ j^{'} (ζ)}{j (ζ)} \in k - P [N, L] (ζ \in U; k ≧ 0) .

Now, we recollect some fundamental definitions and concepts of the q-calculus, which will be used in this paper. Unless we mention otherwise, we assume that

0 < q < 1

and

k \in N \cup \{0\} .

Definition 6.

Let q be within

(0, 1)

. The q-number

{[λ]}_{q}

is defined by

{[λ]}_{q} = \frac{1 - q^{λ}}{1 - q} (λ \in C) .

The q-factorial

{[n]}_{q}!

is defined by

{[n]}_{q}! = \{\begin{matrix} 1 & (n = 0) \\ \prod_{k = 1}^{n} {[k]}_{q} & (n \in N) . \end{matrix}

In particular, if

λ = n \in N,

then

{[n]}_{q} = \sum_{k = 0}^{n - 1} q^{k} = 1 + q + q^{2} + \dots + q^{n - 1} .

Definition 7

([12,13]). The q-derivative (or q-difference) operator

D_{q}

of a function j in a subset of

C

is defined by

(D_{q} j) (ζ) = \{\begin{matrix} \frac{j (ζ) - j (q ζ)}{(1 - q) ζ} & (ζ \neq 0) \\ j^{'} (0) & (ζ = 0), \end{matrix}

(11)

j^{'} (0)

exists.

For a differentiable function j in a subset of

C

, we can see from Definition 7 that

lim_{q \to 1 -} (D_{q} j) (ζ) = lim_{q \to 1 -} \frac{j (ζ) - j (q ζ)}{(1 - q) ζ} = j^{'} (ζ) .

Also, the Equations (1) and (11) give

(D_{q} j) (ζ) = 1 + \sum_{n = 2}^{\infty} {[n]}_{q} a_{n} ζ^{n - 1} .

(12)

Furthermore, one can see from (1) and (11) that

(D_{q}^{(1)} j) (ζ) = {[p]}_{q} ζ^{p - 1} + \sum_{n = 1}^{\infty} {[n + p]}_{q} a_{n + p} ζ^{n + p - 1},

(13)

(D_{q}^{(2)} j) (ζ) = {[p]}_{q} {[p - 1]}_{q} ζ^{p - 2} + \sum_{n = 1}^{\infty} {[n + p]}_{q} {[n + p - 1]}_{q} a_{n + p} ζ^{n + p - 2} .

(14)

Differentiating p times, we obtain

(D_{q}^{(p)} j) (ζ) = {[p]}_{q}! + \sum_{n = 1}^{\infty} \frac{{[n + p]}_{q}!}{{[n]}_{q}!} a_{n + p} ζ^{n},

(15)

where

(D_{q}^{(p)} j) (ζ)

is the p-th time q-deravative of

j (ζ)

In geometric function theory, many subclasses of class

A

of normalized analytic functions have already been discussed within a contrasting frame of reference. The q-calculus provides a key instrument to explore the subclasses of the normalized analytic functions in

A

. Historically, the class of q-starlike functions was first studied by Ismail et al. [14]. However, Srivastava ([15] p. 347 et seq.—also see [1,16,17])—was the first who used the q-calculus to develop some relations between hypergeometric functions and geometric function theory. Some interesting recent developments in this area can be found in [18]. Later, inspired by the prior work, extensive explorations played a key role in the development. For example, the q-analogue of Ruscheweyh’s derivative operator was introduced in [8]. For some recent investigations on this subject, the readers may see [19,20,21,22,23,24].

The following notation

S_{q}^{*}

was first used by Sahoo and Sherma [25].

Definition 8

([14,15]). A function

j \in A

is said to be class

S_{q}^{*}

|\frac{ζ}{j (ζ)} (D_{q} j) (ζ) - \frac{1}{1 - q}| ≦ \frac{1}{1 - q} .

(16)

Equivalently ([26]),

\frac{ζ}{j (ζ)} (D_{q} j) (ζ) ≺ \hat{φ} (ζ) (\hat{φ} (ζ) = \frac{1 + ζ}{1 - q ζ}) .

Now, making use of the principle of subordination between analytic functions and the above-mentioned q-calculus, we define the class k-

P_{q}

Definition 9.

A function φ is said to be in class k-

P_{q}

φ (ζ) ≺ {\hat{φ}}_{k} (ζ)

(17)

where

φ_{k} (ζ)

is defined by (6) and

{\hat{φ}}_{k} (ζ)

is by

{\hat{φ}}_{k} (ζ) = \frac{2 φ_{k} (ζ)}{(1 + q) + (1 - q) φ_{k} (ζ)} .

(18)

Geometrically, the function

φ \in k

P_{q}

takes all points in the domain

Ω_{k, q}

, which is defined as follows ([21,23,27]):

Ω_{k, q} = \{w : ℜ (\frac{(1 + q) w}{(q - 1) w + 2}) > k |\frac{(1 + q) w}{(q - 1) w + 2} - 1|\} .

Domain

Ω_{k, q}

represents a generalized conic region.

One way to generalize the class k-

P [N, L]

in Definition 5 is to replace the function

φ_{k} (ζ)

in (10) by the function

{\hat{φ}}_{k} (ζ),

which is involved in (17). The appropriate definition of the corresponding q-extension of class k-

P [N, L]

is given below.

Definition 10.

A function

h \in P

is said to be class

P (q, k, N, L)

h (ζ) ≺ \frac{(N + 1) {\hat{φ}}_{k} (ζ) - (N - 1)}{(L + 1)) {\hat{φ}}_{k} (ζ) - (L - 1)} (- 1 ≦ L < N ≦ 1; k ≧ 0),

where

{\hat{φ}}_{k} (ζ)

is given by (18) and

φ_{k} (ζ)

is by (6).

Geometrically, the function

h \in P (q, k, N, L)

takes on all points in the domain

Ω_{k} [q, N, L],

which is defined as follows, and represents a generalized conic-type region:

\begin{matrix} Ω_{k} [q, N, L] & = \{w : ℜ (\frac{(1 + q) \{(L - 1) w - (N - 1)\}}{\{(L + 3) + q (L - 1)\} w - \{(N + 3) + q (N - 1)\}}) \\ > k |\frac{(1 + q) \{(L - 1) w - (N - 1)\}}{\{(L + 3) + q (L - 1)\} w - \{(N + 3) + q (N - 1)\}} - 1|\} . \end{matrix}

As an application of Definition 10, we introduce and study the corresponding q-extension of class k-

S^{*} [N, L],

which involves higher-order q-derivatives, below.

Definition 11.

A function

j \in A (p)

is said to be class

S^{*} (q, k, N, L, p)

ℜ (F (q, k, N, L, p)) > k |F (q, k, N, L, p) - 1|,

where

F (q, k, N, L, p) = \frac{(1 + q) \{(L - 1) (\frac{ζ (D_{q}^{(p)} j) (ζ)}{(D_{q}^{(p - 1)} j) (ζ)}) - (N - 1)\}}{\{(L + 3) + q (L - 1)\} (\frac{ζ (D_{q}^{(p)} j) (ζ)}{(D_{q}^{(p - 1)} j) (ζ)}) - \{(N + 3) + q (N - 1)\}} .

(19)

Equivalently,

\frac{ζ (D_{q} j) (ζ)}{j (ζ)} \in P (q, k, N, L) .

(20)

It is worthwhile to note the following special subclasses of class

S^{*} (q, k, N, L)

(a) For

k = 0, N = 1 - 2 α (0 \leq α < 1), L = - 1 = - p,

and

q \to 1 - :

it gives class

S^{*} (α)

(see [28]).

(b) For

N = 1 = p, L = - 1,

and

q \to 1 - :

it gives class k-

ST

(see [5]).

N = 1 - 2 α (0 \leq α < 1) = - 1 = - p,

and

q \to 1 - :

it gives class

SD (k, α)

(see [10]).

(d) For

k = 0, N = 1 = p L = - 1,

and

q \to 1 - :

it gives class

S_{q}^{*}

(see [14]).

(e) For

p = 1

and

q \to 1 -

: it gives class k-

S^{*} [N, L]

(see [11]).

(f) For

k = 0 = p - 1 :

it gives class

S_{q}^{*} [N, L]

(see [22]).

Lemma 1

([29,30]). Let

Φ (ζ) = 1 + c_{1} ζ + c_{2} ζ^{2} + \dots

be in class

P

; then, for any complex number

υ,

|c_{2} - υ c_{1}^{2}| ≦ 2 max \{1, |1 - 2 υ|\} .

If υ is a real parameter, by Lemma then

|c_{2} - υ c_{1}^{2}| ≦ \{\begin{matrix} - 4 υ + 2 & (υ ≦ 0) \\ 2 & (0 ≦ υ ≦ 1) \\ 4 υ - 2 & (υ ≧ 1) . \end{matrix}

(21)

2. A Set of Main Results

Theorem 1.

A function

j \in A (p)

of the form given by (1) is in class

S^{*} (q, k, N, L, p)

if it satisfies the following condition

\sum_{n = 1}^{\infty} Λ (n, k, N, L, q, p) |a_{n + p}| < |L - N| (1 + q),

(22)

where

Λ (n, k, N, L, q, p) = 4 (k + 1) q {[n]}_{q} + |L (n, k, N, L, q, p)|

(23)

and

L (n, k, N, L, q, p) = \{L (1 + q) + 3 - q\} \frac{{[n + p]}_{q}! {[n + 1]}_{q}}{{[n + 1]}_{q}!} - \{N (q + 1) + 3 - q\} .

(24)

Proof.

Assuming that (22) holds, by Definition 11, it suffices to show that

k |F (q, k, N, L, p) - 1| - ℜ (F (q, k, N, L, p) - 1) < 1,

where

F (q, k, N, L)

is given by (19).

Now, we have

\begin{matrix} (k + 1) |F (q, k, N, L, p) - 1| \\ \leq (k + 1) |\frac{(1 + q) \{(L - 1) ζ (D_{q}^{(p)} j) (ζ) - (N - 1) (D_{q}^{(p - 1)} j) (ζ)\}}{M (N, L, k, q, p)} - 1| \\ = 4 (k + 1) |\frac{(D_{q}^{(p - 1)} j) (ζ) - ζ (D_{q}^{(p)} j) (ζ)}{M (N, L, k, q, p)}| \\ = 4 (k + 1) |\frac{\sum_{n = 1}^{\infty} \frac{{[n + p]}_{q}!}{{[n + 1]}_{q}!} (1 - {[n + 1]}_{q}) a_{n + p} ζ^{n + 1}}{(L - N) (1 + q) {[p]}_{q}! ζ + \sum_{n = 1}^{\infty} L (n, k, N, L, q, p) a_{n + p} ζ^{n + 1}}| \\ \leq \frac{4 (k + 1) \sum_{n = 1}^{\infty} \frac{{[n + p]}_{q}!}{{[n + 1]}_{q}!} (1 - {[1 + n]}_{q}) |a_{n + p}|}{|(L - N) (1 + q)| {[p]}_{q}! - \sum_{n = 2}^{\infty} L (n, k, N, L, q, p) |a_{n + p}|}, \end{matrix}

(25)

where

M (N, L, k, q, p) = \{(L + 3) + q (L - 1)\} ζ (D_{q}^{(p)} j) (ζ) - \{(N + 3) + q (N - 1)\} D_{q}^{(p - 1)} j (ζ)

and

L (n, k, N, L, q, p)

is given by (24).

If (22) holds,

\sum_{n = 1}^{\infty} Λ (n, k, N, L, q, p) |a_{n + p}| < |L - N| (1 + q) .

Using the identities (23) and (24), the last expression in (25) is bounded above by 1, and hence the proof is now completed. □

Upon letting

q \to 1 -,

and

p = 1,

Theorem 1 yields the following known result.

Corollary 1

([11]). A function

j \in A

of the form given by (1) is in class k-

S^{*} [N, L]

if it satisfies the following condition

\sum_{n = 2}^{\infty} \{2 n (k + 1) + |(n + 1) (L + 1) - (N + 1)|\} |a_{n}| < |L - N| .

In Theorem 1, if we set

k = 0, N = 1 - 2 α (0 \leq α < 1) and L = - 1 = - p,

and let

q \to 1 -

, we are led to the following known result.

Corollary 2

([28]). A function

j \in A

of the form given by (1) is in class

S^{*} (α)

if it satisfies the condition

\sum_{n = 2}^{\infty} (n + α) |a_{n}| < 1 - α (0 \leq α < 1) .

Furthermore, if we put

p = N = 1 and L = - 1,

and let

q \to 1 -

, then Theorem 1 implies the following corollary.

Corollary 3

([5]). A function

j \in A

of the form given by (1) is in class

k - ST

if it satisfies the condition

\sum_{n = 2}^{\infty} \{n (k + 1) + 1\} |a_{n}| < 1 .

Moreover, if we put

N = 1 - 2 α (0 \leq α < 1) and L = - 1 = - p,

and let

q \to 1 -

, then Theorem 1 implies the following known result.

Corollary 4

([10]). A function

j \in A

of the form given by (1) is in class

SD (k, α)

if it satisfies the condition

\sum_{n = 2}^{\infty} \{n (k + 1) + α\} |a_{n}| < (1 - α) .

Theorem 2.

Let

j (ζ)

be a function with (1) in class

S^{*} (q, k, N, L, p),

where

0 \leq k \leq 1

. (a) If μ is a complex number, then

\begin{matrix} |a_{p + 2} - μ a_{p + 1}^{2}| & \leq & (\frac{(1 - q^{2}) (1 - q^{3})}{(1 - q^{p + 1}) (1 - q^{p + 2})}) (\frac{N - L}{8 q}) P_{1} \\ \cdot max \{1, |\frac{P_{2}}{P_{1}} + \frac{Υ (q)}{4 q} P_{1} - \frac{μ (N - L) {(1 + q)}^{2}}{4 q} P_{1}|\} . \end{matrix}

(26)

(b) If μ is a real parameter, then

|a_{p + 2} - μ a_{p + 1}^{2}| \leq \{\begin{matrix} (\frac{(1 - q^{2}) (1 - q^{3})}{(1 - q^{p + 1}) (1 - q^{p + 2})}) (\frac{N - L}{4 q}) (P_{2} + \frac{Υ (q)}{4 q} P_{1}^{2} - \frac{μ {(1 + q)}^{2}}{4 q} P_{1}^{2}) & (μ < σ_{1}) \\ (\frac{(1 - q^{2}) (1 - q^{3})}{(1 - q^{p + 1}) (1 - q^{p + 2})}) (\frac{N - L}{4 q}) P_{1} & (σ_{1} \leq μ \leq σ_{2}) \\ (\frac{(1 - q^{2}) (1 - q^{3})}{(1 - q^{p + 1}) (1 - q^{p + 2})}) (\frac{L - N}{4 q}) (P_{2} + \frac{Υ (q)}{4 q} P_{1}^{2} - \frac{μ {(1 + q)}^{2}}{4 q} P_{1}^{2}) & (μ > σ_{2}), \end{matrix}

(27)

where

Υ (q) = [(N - L) + (N - 5 L - 3) q + (1 - L) q^{2}],

(28)

σ_{1} = \frac{4 q {[p + 1]}_{q} {[3]}_{q}}{(N - L) {(1 + q)}^{2} {[2]}_{q} {[p + 2]}_{q} P_{1}^{2}} \{\frac{Υ (q)}{4 q} P_{1}^{2} - P_{1} + P_{2}\}

σ_{2} = \frac{4 q {[p + 1]}_{q} {[3]}_{q}}{(N - L) {(1 + q)}^{2} {[2]}_{q} {[p + 2]}_{q} P_{1}^{2}} \{P_{1} + P_{2} + \frac{Υ (q)}{4 q} P_{1}^{2}\}

where

P_{1}, P_{2}

are defined by (7), (8), respectively. The result is sharp for a real parameter μ.

Proof.

We begin by showing that the inequalities (26) and (27) are true for

j \in S^{*} (q, k, N, L, p)

. Let us consider a function

Ψ (ζ)

given by

Ψ (ζ) = \frac{ζ (D_{q}^{(p)} j) (ζ)}{D^{(p - 1)} j (ζ)} (\forall ζ \in U) .

Since

j \in S^{*} (q, k, N, L, p),

we have the following subordination:

Ψ (ζ) ≺ ϕ (ζ),

(29)

where

ϕ (ζ) = \frac{(1 + q) (N + 1) (p_{k} (ζ) - 1) + 2 (p_{k} (ζ) + 1 - q (p_{k} (ζ) - 1))}{(1 + q) (L + 1) (p_{k} (ζ) - 1) + 2 (p_{k} (ζ) + 1 - q (p_{k} (ζ) - 1))} .

Suppose that

p_{k} (ζ) = 1 + P_{1} ζ + P_{2} ζ^{2} + . . .,

then we can find, after some simplification, that

\begin{matrix} ϕ (ζ) & = 1 + \frac{1}{4} (N - L) (q + 1) P_{1} ζ + \frac{1}{16} (N - L) (q + 1) \\ \cdot \{4 P_{2} - (3 - q + (q + 1) L) P_{1}^{2}\} ζ^{2} + . . . \end{matrix}

Using the subordination relation (29), we see that the function

h (ζ)

given by

h (ζ) = \frac{1 + ϕ^{- 1} (Ψ (ζ))}{1 - ϕ^{- 1} (Ψ (ζ))} = 1 + c_{1} ζ + c_{2} ζ^{2} + . . . (\forall ζ \in U)

belongs to the class

P

. We also have

Ψ (ζ) = ϕ (\frac{h (ζ) - 1}{h (ζ) + 1}) .

(30)

where

\begin{matrix} Ψ (ζ) & = & \frac{ζ (D_{q}^{(p)} j) (ζ)}{(D_{q}^{(p - 1)} j) (ζ)} = 1 + \frac{q}{1 + q} (\frac{1 - q^{p + 1}}{1 - q}) a_{p + 1} ζ \\ + \{\frac{q (1 - q^{p + 1}) (1 - q^{p + 2})}{(1 - q) (1 - q^{3})} a_{p + 2} - \frac{q {(1 - q^{p + 1})}^{2}}{{(1 - q)}^{2} {(1 + q)}^{2}} a_{p + 1}^{2}\} ζ^{2} \\ + . . . \end{matrix}

(31)

and

\begin{matrix} ϕ (\frac{Φ (ζ) - 1}{Φ (ζ) + 1}) & = 1 + \frac{1}{8} (N - L) (q + 1) P_{1} c_{1} ζ + \frac{1}{8} (N - L) (q + 1) \\ \cdot \{P_{1} c_{2} + (\frac{P_{2}}{2} - \frac{(3 - q + (q + 1) L)}{8} P_{1}^{2} - \frac{P_{2}}{2}) c_{1}^{2}\} ζ^{2} + . . . . \end{matrix}

(32)

From Equations (31) and (32),

a_{p + 1} = \frac{(N - L) {(q + 1)}^{2}}{8 q {[p + 1]}_{q}} P_{1} c_{1}

(33)

and

a_{p + 2} = \frac{(N - L) {[2]}_{q} {[3]}_{q}}{8 q {[p + 1]}_{q} {[p + 2]}_{q}} [P_{1} c_{2} + (\frac{P_{2}}{2} - \frac{P_{1}}{2} + \frac{Υ (q) P_{1}^{2}}{8 q}) c_{1}^{2}],

(34)

where

Υ (q)

is given by (28).

Thus,

|a_{p + 2} - μ a_{p + 1}^{2}| = (\frac{(N - L) {[2]}_{q} {[3]}_{q}}{8 q {[p + 1]}_{q} {[p + 2]}_{q}}) P_{1} |c_{2} - ζ c_{1}^{2}|,

(35)

where

ζ = \frac{1}{2} (1 - \frac{P_{2}}{P_{1}} - \frac{Υ (q) P_{1}}{4 q} + \frac{μ (N - L) {(1 + q)}^{2} {[2]}_{q} {[p + 2]}_{q} P_{1}}{4 q {[p + 1]}_{q} {[3]}_{q}}) .

Now applying for Lemma 1 with (35), we obtain the results stated in (27). □

3. Partial Sums for the Function Class $S^{*} (q, k, N, L, p)$

We assume that j is of the form (1) unless otherwise stated. Thus, its sequence of partial sums is given by

j_{m} (ζ) = ζ^{p} + \sum_{n = 1}^{m} a_{n + p} ζ^{n + p} .

we then examine the ratio of a function j to its sequence of partial sums.

Also, we obtain the sharp lower bounds for

ℜ (\frac{j (ζ)}{j_{m} (ζ)}), (\frac{j_{m} (ζ)}{j (ζ)}), ℜ (\frac{D_{q}^{(p)} j (ζ)}{D_{q}^{(p)} j_{m} (ζ)}) and ℜ (\frac{(D_{q}^{(p)} j_{m}) (ζ)}{(D_{q}^{(p)} j) (ζ)}) .

Theorem 3.

If j of the form (1) satisfies condition (22), then

ℜ (\frac{j (ζ)}{j_{m} (ζ)}) \geq 1 - \frac{1}{ρ_{j + 1}} (ζ \in U)

(36)

and

ℜ (\frac{j_{m} (ζ)}{j (ζ)}) \geq \frac{ρ_{j + 1}}{1 + ρ_{j + 1}} (ζ \in U),

(37)

where

ρ_{m} = \frac{Λ (j, k, N, L, q, p)}{(1 + q) |N - L|}

(38)

and

Λ (j, k, N, L, q, p)

is given by (23).

Proof.

In order to prove the inequality (36), we set

\begin{matrix} ρ_{j + 1} [\frac{j (ζ)}{j_{m} (ζ)} - (1 - \frac{1}{ρ_{j + 1}})] & = \frac{1 + \sum_{n = 1}^{m} a_{n + p} ζ^{n} + ρ_{j + 1} \sum_{n = j + 1}^{\infty} a_{n + p} ζ^{n}}{1 + \sum_{n = 2}^{m} a_{n + p} ζ^{n}} \\ = \frac{1 + h_{1} (ζ)}{1 + h_{2} (ζ)} . \end{matrix}

If we set,

\frac{1 + h_{1} (ζ)}{1 + h_{2} (ζ)} = \frac{1 + w (ζ)}{1 - w (ζ)},

then we have,

w (ζ) = \frac{h_{1} (ζ) - h_{2} (ζ)}{2 + h_{1} (ζ) + h_{2} (ζ)} .

It follows that

w (ζ) = \frac{ρ_{j + 1} \sum_{n = j + 1}^{\infty} a_{n + p} ζ^{n}}{2 + 2 \sum_{n = 1}^{m} a_{n + p} ζ^{n} + ρ_{j + 1} \sum_{n = j + 1}^{\infty} a_{n + p} ζ^{n}}

and

|w (ζ)| \leq \frac{ρ_{j + 1} \sum_{n = j + 1}^{\infty} |a_{n + p}|}{2 - 2 \sum_{n = 1}^{m} |a_{n + p}| - ρ_{j + 1} \sum_{n = j + 1}^{\infty} |a_{n + p}|} .

It is not difficult to see that

|w (ζ)| \leq 1

is equivalent to

2 ρ_{j + 1} \sum_{n = j + 1}^{\infty} |a_{n + p}| \leq 2 - 2 \sum_{n = 1}^{m} |a_{n + p}|,

which implies that

\sum_{n = 1}^{m} |a_{n + p}| + ρ_{j + 1} \sum_{n = j + 1}^{\infty} |a_{n + p}| \leq 1 .

(39)

To prove our assertion (36), it is enough to show that (39) is bounded above by

\sum_{n = 2}^{\infty} ρ_{n} |a_{n}|,

which can be written as

\sum_{n = 1}^{m} (1 - ρ_{n}) |a_{n + p}| + \sum_{n = j + 1}^{\infty} (ρ_{j + 1} - ρ_{n}) |a_{n + p}| \geq 0 .

(40)

Thus, the proof of inequality in (36) is completed now.

Next, in order to prove the inequality (37), we set

\begin{matrix} (1 + ρ_{m}) (\frac{j_{m} (ζ)}{j (ζ)} - \frac{ρ_{m}}{1 + ρ_{m}}) & = \frac{1 + \sum_{n = 1}^{m} a_{n + p} ζ^{n} - ρ_{j + 1} \sum_{n = j + 1}^{\infty} a_{n + p} ζ^{n}}{1 + \sum_{n = 1}^{\infty} a_{n + p} ζ^{n}} \\ = \frac{1 + w (ζ)}{1 - w (ζ)}, \end{matrix}

where

|w (ζ)| \leq \frac{(1 + ρ_{j + 1}) \sum_{n = j + 1}^{\infty} |a_{n + p}|}{2 - 2 \sum_{n = 1}^{m} |a_{n + p}| - (ρ_{j + 1} - 1) \sum_{n = j + 1}^{\infty} |a_{n + p}|} \leq 1 .

(41)

The right-hand side inequality in (42) is equivalent to

\sum_{n = 1}^{m} |a_{n + p}| + ρ_{j + 1} \sum_{n = j + 1}^{\infty} |a_{n + p}| \leq 1 .

(42)

Also, the left-hand side inequality in (42) is bounded above by

\sum_{n = 2}^{\infty} ρ_{n} |a_{n + p}|

and so we have completed the proof of (37). This completes the proof of Theorem 3. □

The following result is related to the functions involving derivatives.

Theorem 4.

If j of the form (1) satisfies condition (22), then

ℜ (\frac{(D_{q}^{(p)} j) (ζ)}{(D_{q}^{(p)} j_{m}) (ζ)}) \geq 1 - \frac{{[j + p]}_{q}}{ρ_{j + 1}} (\forall ζ \in U)

(43)

and

ℜ (\frac{(D_{q}^{(p)} j_{m}) (ζ)}{(D_{q}^{(p)} j) (ζ)}) \geq \frac{ρ_{j + 1}}{ρ_{j + 1} + {[j + p]}_{q}} (\forall ζ \in U),

(44)

where

ρ_{m}

is given by (38).

Proof.

The proof of Theorem 4 is similar to that of Theorem 3; we here choose to omit the analogous details. □

4. Analytic Functions with Negative Coefficients

First, we give a new subclass of starlike functions having negative coefficients. Let

T \subset A,

which consists of functions having negative coefficients, i.e.,

j (ζ) = ζ^{p} - \sum_{n = 1}^{\infty} |a_{n + p}| ζ^{n + p} .

(45)

Now, we state and prove the distortion results for subclass

{TS}^{*} (N, L, q, k, p)

T .

Theorem 5.

j \in {TS}^{*} (N, L, q, k, p),

then

r^{p} - \frac{|L - N| (1 + q)}{Λ (2, k, N, L, q, p)} r^{1 + p} \leq |j (ζ)| \leq r^{p} + \frac{|L - N| (1 + q)}{Λ (2, k, N, L, q, p)} r^{1 + p}, (|ζ| = r)

where

Λ (2, k, N, L, q, p)

is given by (23).

Proof.

We note that the following inequality follows from Theorem 1:

Λ (2, k, N, L, q, p) \sum_{n = 1}^{\infty} |a_{n + p}| \leq \sum_{n = 1}^{\infty} Λ (n, k, N, L, q, p) |a_{n + p}| < |L - N| (1 + q),

which yields

|j (ζ)| \leq r^{p} + \sum_{n = 1}^{\infty} |a_{n + p}| r^{n + p} \leq r^{p} + r^{1 + p} \sum_{n = 1}^{\infty} |a_{n + p}| \leq r^{p} + \frac{|L - N| (1 + q)}{Λ (2, k, N, L, q, p)} r^{1 + p},

Also, we have

|j (ζ)| \geq r^{p} - \sum_{n = 1}^{\infty} |a_{n + p}| r^{n + p} \geq r^{p} - r^{1 + p} \sum_{n = 1}^{\infty} |a_{n + p}| \geq r^{p} - \frac{|L - N| (1 + q)}{Λ (2, k, N, L, q, p)} r^{1 + p} .

Therefore, the proof is completed. □

In the special case that

k = 0, N = 1 - 2 α (0 \leq α < 1) and L = - 1, p = 1,

and if we let

q \to 1 -

, Theorem 5 reduces to the following known result.

Corollary 5

([28]). If

j \in {TS}^{*} (α),

then

r - \frac{1 - α}{2 - α} r^{2} \leq |j (ζ)| \leq r + \frac{1 - α}{2 - α} r^{2}, (|ζ| = r) .

The proof of the following theorem is similar to the proof of Theorem 5; therefore, it is omitted.

Theorem 6.

j \in {TS}^{*} (N, L, q, k, p),

then

1 - \frac{2 |L - N|}{Λ (2, k, N, L, q, p)} r^{p} \leq |j^{'} (ζ)| \leq 1 + \frac{2 |L - N|}{Λ (2, k, N, L, q, p)} r^{p}, (|ζ| = r)

where

Λ (2, k, N, L, q, p)

is given by (23).

In the special case that

k = 0, N = 1 - 2 α, (0 \leq α < 1) L = - 1, and p = 1,

and if we let

q \to 1 -

, Theorem 6 reduces to the known result given in [28].

5. Concluding Remarks and Observations

In this paper, we have systematically used the conic domain and the celebrated Janowski functions with higher-order q-derivatives. We have defined a new subclass of q-starlike functions. We have then obtained several remarkable results, such as sufficient conditions and some coefficient estimates. We have also given some specific cases of our main results in the form of remarks and corollaries.

Author Contributions

All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

Funding

United Arab Emirates University, UPAR 31S315.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

Aldweby, H.; Darus, M. Some subordination results on q-analogue of Ruscheweyh differential operator. Abstr. Appl. Anal. 2014, 2014, 958563. [Google Scholar] [CrossRef]
Cho, N.E.; Srivastava, H.M.; Adegani, E.A.; Motamednezhad, A. Criteria for a certain class of the Carathéodory functions and their applications. J. Inequal. Appl. 2020, 2020, 85. [Google Scholar] [CrossRef]
Janowski, W. Some extremal problems for certain families of analytic functions. Ann. Polon. Math. 1973, 28, 297–326. [Google Scholar] [CrossRef]
Kanas, S.; Wiśniowska, A. Conic regions and k-uniform convexity. J. Comput. Appl. Math. 1999, 105, 327–336. [Google Scholar] [CrossRef]
Kanas, S.; Wiśniowska, A. Conic domains and starlike functions. Rev. Roumaine Math. Pures Appl. 2000, 45, 647–657. [Google Scholar]
Kanas, S.; Srivastava, H.M. Linear operators associated with k-uniformly convex functions. Integral Transform. Spec. Funct. 2000, 9, 121–132. [Google Scholar] [CrossRef]
Kanas, S. Coefficient estimates in subclasses of the Carathéodary class related to conic domains. Acta Math. Univ. Comen. 2005, 74, 149–161. [Google Scholar]
Kanas, S.; Răducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
Khan, N.; Khan, B.; Ahmad, Q.Z.; Ahmad, S. Some Convolution properties of multivalent analytic functions. AIMS Math. 2017, 2, 260–268. [Google Scholar] [CrossRef]
Shams, S.; Kulkarni, S.R.; Jahangiri, J.M. Classes of uniformly starlike and convex functions. Int. J. Math. Math. Sci. 2004, 55, 2959–2961. [Google Scholar] [CrossRef]
Noor, K.I.; Malik, S.N. On coefficient inequalities of functions associated with conic domains. Comput. Math. Appl. 2011, 62, 2209–2217. [Google Scholar] [CrossRef]
Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
Jackson, F.H. q-difference equations. Am. J. Math. 1910, 32, 305–314. [Google Scholar] [CrossRef]
Ismail, M.E.-H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [Google Scholar] [CrossRef]
Srivastava, H.M.; Owa, S. (Eds.) Univalent Functions, Fractional Calculus, and Associated Generalized Hypergeometric Functions; Ellis Horwood Limited: Chichester, UK, 1989. [Google Scholar]
Ezeafulukwe, U.A.; Darus, M. Certain properties of q-hypergeometric functions. Int. J. Math. Math. Sci. 2015, 2015, 489218. [Google Scholar] [CrossRef]
Huda, A.; Darus, M. Partial sum of generalized class of meromorphically univalent functions defined by q-analogue of Liu-Srivastava operator. Asian-Eur. J. Math. 2014, 7, 1450046. [Google Scholar]
Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 2020, 44, 327–344. [Google Scholar] [CrossRef]
Mahmood, S.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M.J. Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative. J. Funct. Spaces 2018, 2018, 8492072. [Google Scholar] [CrossRef]
Mahmood, S.; Srivastava, H.M.; Khan, N.; Ahmad, Q.Z.; Khan, B.; Ali, I. Upper bound of the third Hankel determinant for a subclass of q-starlike functions. Symmetry 2019, 11, 347. [Google Scholar] [CrossRef]
Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B. Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 2019, 7, 181. [Google Scholar] [CrossRef]
Srivastava, H.M.; Khan, B.; Khan, N.; Ahmad, Q.Z. Coefficient inequalities for q-starlike functions associated with the Janowski functions. Hokkaido Math. J. 2019, 48, 407–425. [Google Scholar] [CrossRef]
Khan, B.; Liu, Z.-G.; Shaba, T.G.; Araci, S.; Khan, N.; Khan, M.G. Applications of q-Derivative Operator to the Subclass of Bi-Univalent Functions Involving q-Chebyshev Polynomials. J. Math. 2022, 2022, 8162182. [Google Scholar] [CrossRef]
Srivastava, H.M.; Tahir, M.; Khan, B.; Ahmad, Q.Z.; Khan, N. Some general classes of q-starlike functions associated with the Janowski functions. Symmetry 2019, 11, 292. [Google Scholar] [CrossRef]
Sahoo, S.K.; Sharma, N.L. On a generalization of close-to-convex functions. Ann. Polon. Math. 2015, 113, 93–108. [Google Scholar] [CrossRef] [Green Version]
Uçar, H.E.Ö. Coefficient inequality for q-starlike functions. Appl. Math. Comput. 2016, 276, 122–126. [Google Scholar]
Rehman, M.S.U.; Ahmad, Q.Z.; Srivastava, H.M.; Khan, N.; Darus, M.; Khan, B. Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions. AIMS Math. 2021, 6, 1110–1125. [Google Scholar] [CrossRef]
Silverman, H. Univalent functions with negative coefficients. Proc. Am. Math. Soc. 1975, 51, 109–116. [Google Scholar] [CrossRef]
Keogh, F.R.; Merkes, E.P. A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969, 20, 8–12. [Google Scholar] [CrossRef]
Ma, W.; Minda, D. A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis, Tianjin, China, 19–23 June 1992; Li, Z., Ren, F., Zhang, L.Y.S., Eds.; International Press Inc.: Cambridge, MA, USA, 1994; pp. 157–169. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Ur Rehman, M.S.; Ahmad, Q.Z.; Al-shbeil, I.; Ahmad, S.; Khan, A.; Khan, B.; Gong, J. Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains. Axioms 2022, 11, 494. https://doi.org/10.3390/axioms11100494

AMA Style

Ur Rehman MS, Ahmad QZ, Al-shbeil I, Ahmad S, Khan A, Khan B, Gong J. Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains. Axioms. 2022; 11(10):494. https://doi.org/10.3390/axioms11100494

Chicago/Turabian Style

Ur Rehman, Muhammad Sabil, Qazi Zahoor Ahmad, Isra Al-shbeil, Sarfraz Ahmad, Ajmal Khan, Bilal Khan, and Jianhua Gong. 2022. "Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains" Axioms 11, no. 10: 494. https://doi.org/10.3390/axioms11100494

APA Style

Ur Rehman, M. S., Ahmad, Q. Z., Al-shbeil, I., Ahmad, S., Khan, A., Khan, B., & Gong, J. (2022). Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains. Axioms, 11(10), 494. https://doi.org/10.3390/axioms11100494

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains

Abstract

1. Introduction and Preliminaries

2. A Set of Main Results

3. Partial Sums for the Function Class $S^{*} (q, k, N, L, p)$

4. Analytic Functions with Negative Coefficients

5. Concluding Remarks and Observations

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Article Menu

Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains

Abstract

1. Introduction and Preliminaries

2. A Set of Main Results

3. Partial Sums for the Function Class S ∗ q , k , N , L , p

4. Analytic Functions with Negative Coefficients

5. Concluding Remarks and Observations

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

3. Partial Sums for the Function Class $S^{*} (q, k, N, L, p)$