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New refinements of some classical inequalities via Young’s inequality

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Abstract

The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every \(T,S \in {\mathcal {B(H)}}\), \(\alpha \in (0,1)\) and \(x, y \in {\mathcal {H}}\), we prove that

$$\begin{aligned}{} & {} \left( 1+ L(\alpha )\log ^2\left( \frac{|\langle TS x, y\rangle | }{r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| }\right) \right) |\langle TSx, y\rangle | \\{} & {} \quad \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}$$

where L is a positive 1-periodic function and r(S) is the spectral radius of S, which gives an improvement of the well-known generalized mixed Schwarz inequality:

$$\begin{aligned} \left| \langle TSx,y \rangle \right| \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}$$

where \(|T| S=S^*|T|\) and f, g are non-negative continuous functions defined on \([0, \infty )\) satisfying that \(f(t) g(t)=t\,(t \ge 0)\).

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Acknowledgements

The authors would like to express their sincere gratitude to the referee for his careful review of this paper and for his valuable comments and suggestions.

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Authors and Affiliations

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

    Mohamed Amine Ighachane

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

    Fuad Kittaneh

  3. Department of Mathematics, Faculy of Sciences-Semlalia, University Cadi Ayyad, Av. Prince My. Abdellah, BP: 2390, 40000, Marrakech, Morocco

    Zakaria Taki

Authors
  1. Mohamed Amine Ighachane
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  2. Fuad Kittaneh
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  3. Zakaria Taki
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Authors declare that they have contributed equally to this paper. All authors have read and approved this version.

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Correspondence to Fuad Kittaneh.

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Communicated by Hiroyuki Osaka.

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Ighachane, M.A., Kittaneh, F. & Taki, Z. New refinements of some classical inequalities via Young’s inequality. Adv. Oper. Theory 9, 49 (2024). https://doi.org/10.1007/s43036-024-00347-4

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