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New improvements of some classical inequalities

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Abstract

In this paper, we establish an inequality for scalars, which we then apply to refine some classical inequalities for inner product and numerical raduis. For example, we establish that for any \(\mathcal {E}\in \mathcal {B}(\mathcal {H}),\) \(u,v\in \mathcal {H},\) and \(0\le \theta \le 1\),

$$\begin{aligned} |\langle \mathcal {E} u,v\rangle |^2&\le \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) \\ &\le \left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle . \end{aligned}$$

Moreover, we have \(\left( \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) \right) _{n \geqslant 0}\) is an increasing sequence satisfying

$$\begin{aligned} \lim \limits _{n \rightarrow +\infty } \mathcal {U}_{(n,\xi )}\left( \eta ,|\langle \mathcal {E}u,v\rangle |,\sqrt{\left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle }\right) = \left\langle |\mathcal {E}|^{2 \theta } u,u\right\rangle \left\langle \left| \mathcal {E}^*\right| ^{2(1-\theta )} v, v\right\rangle , \end{aligned}$$

which presents a novel refinement of the well-known mixed Schwartz inequality. Our results extend and refine well-established inequalities found in the literature.

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Acknowledgements

The authors are grateful to the referees for their useful comments.

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The authors did not receive any financial support for this work.

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Authors and Affiliations

  1. Mathematics and Applications Laboratory Faculty of Sciences of Agadir (FSA), Ibn zohr university, Agadir, Morocco

    Abdelmajid Gourty

  2. Sciences and Technologies Team (ESTE) Higher School of Education and Training of El Jadida, Chouaib Doukkali University, El Jadida, Morocco

    Mohamed Amine Ighachane

  3. Department of Mathematics, The University of Jordan, Amman, Jordan

    Fuad Kittaneh

  4. Department of Mathematics, Korea University, Seoul, 02841, South Korea

    Fuad Kittaneh

Authors
  1. Abdelmajid Gourty
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  2. Mohamed Amine Ighachane
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  3. Fuad Kittaneh
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Correspondence to Fuad Kittaneh.

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Gourty, A., Ighachane, M.A. & Kittaneh, F. New improvements of some classical inequalities. Afr. Mat. 35, 76 (2024). https://doi.org/10.1007/s13370-024-01218-0

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