Search Results (91)

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class $S{T}_q(m,c,d,\beta )$. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties. Full article

Applying the operator of $\mathfrak{q}$-difference, we examine the convolution properties of the subclasses ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{A},\mathcal{B})$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{A},\mathcal{B})$ of p-valent meromorphic functions defined in the punctured open-unit disc. We derived specific inclusion features and coefficient estimates for functions that fall into these subclasses. Additionally, connections between the results presented here and those discovered in earlier papers are emphasized. Full article

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points. Full article

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates $|a_3-\mu a_2^2|$ for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. Full article

The topic concerning the introduction and investigation of new classes of analytic functions using subordination techniques for obtaining certain geometric properties alongside coefficient estimates and inclusion relations is enriched by the results of the present investigation. The prolific tools of quantum calculus applied in geometric function theory are employed for the investigation of a new class of analytic functions introduced by applying a previously defined generalized $\mathfrak{q}-$integral operator and the concept of subordination. Investigations are conducted on the new class, including coefficient estimates, integral representation for the functions of the class, linear combinations, forms of the weighted and arithmetic means involving functions from the class, and the estimation of the integral means results. Full article

Certain characteristics of univalent functions with negative coefficients of the form $f\left(z\right)=z-{\sum }_{n=1}^{\infty }{a}_{2n}{z}^{2n},a_{2n}>0$ have been studied by Silverman and Berman. Pokley, Patil and Shrigan have discovered some insights into the Hadamard product of $\mathcal{P}$-valent functions with negative coefficients. S. M. Khairnar and Meena More have obtained coefficient limits and convolution results for univalent functions lacking a alternating type coefficient. In this paper, using the $\mathfrak{q}$-Difference operator, we developed the a subclass of meromorphically $\mathcal{P}$-valent functions with alternating coefficients. Additionally, we obtained multivalent function convolution results and coefficient limits. Full article

In this article, we first consider the fractional q-differential operator and the $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$. Using the $\left(\lambda ,q\right)$-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ of starlike functions of order $\beta$ and the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions of order $\beta$. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order $\beta$. We also investigate the erratic behavior of the initial coefficients in the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research. Full article

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with $\mathit{q}$-calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on $|{\alpha }_2|$ and $|{\alpha }_3|$ coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. Full article

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies. Full article

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with $\mathit{q}$-calculus. The class is proved to not be empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on $|{\alpha }_2|$ and $|{\alpha }_3|$ coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. Full article

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator $D_{q,\lambda }^m\left(\rho ,\sigma \right)$ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc $\mathcal{U}$. Furthermore, we analyze novel findings with respect to the inverse function $\left({\mu }^{-1}\right)$ within the class of q-starlike functions in $\mathcal{U}$. The findings in this paper are easy to understand and show a connection between present and past studies. Full article

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. Full article

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since $\Re (1+sinh(z\left)\right)\ngtr 0,$ it implies that the class $S_{sinh}^{*}$ introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter $\lambda$ with the restriction $0\le \lambda \le ln(1+\sqrt{2}),$ and by doing that, $\Re (1+sinh(\lambda z\left)\right)>0.$ The present research intends to provide a novel subclass of starlike functions in the open unit disk $\mathcal{U},$ denoted as $S_{sinh\lambda }^{*}$, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients $a_n$ for $n=2,3,4,5.$ Then, we prove a lemma, in which the largest disk contained in the image domain of $q_0\left(z\right)=1+sinh\left(\lambda z\right)$ and the smallest disk containing $q_0\left(\mathcal{U}\right)$ are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including $S^{*}\left(\beta \right)$ and $\mathcal{K}\left(\beta \right)$ of starlike functions of order $\beta$ and convex functions of order $\beta$. Investigating $S_{sinh\lambda }^{*}$ radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding $S_{sinh\lambda }^{*}$ radii of different subclasses is the calculation of that value of the radius $r<1$ for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of $\lambda$, is also obtained. Full article

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution conservation, and convexity constraints. Next, using sufficiency criteria, we calculate the sharp bounds of the real parts of the ratios of harmonic functions to their sequences of partial sums. In addition, for the first time some of the interesting implications of the q-Srivastava–Attiya operator in harmonic functions are also included. Full article

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions. Full article