# Norm Inequalities via Convex and Log-Convex Functions

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## Abstract

In this paper, we study the norm and skew angular distances in a normed space $${\mathscr {X}}$$, where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. For example, in this regard, we show that if $$a,b\in {\mathscr {X}}$$ are non-zero and if $$p,q>0$$ are such that $$\frac{1}{p}+\frac{1}{q}=1$$, then

\begin{aligned} \begin{aligned} 2\lambda \left( \frac{{{p}^{r}}{{\left\| a \right\| }^{r}}+{{q}^{r}}{{\left\| b \right\| }^{r}}}{2}-{{\left\| \frac{pa+qb}{2} \right\| }^{r}} \right)&\le {{p}^{r-1}}{{\left\| a \right\| }^{r}}+{{q}^{r-1}}{{\left\| b \right\| }^{r}}-{{\left\| a+b \right\| }^{r}} \\&\le 2\mu \left( \frac{{{p}^{r}}{{\left\| a \right\| }^{r}}+{{q}^{r}}{{\left\| b \right\| }^{r}}}{2}-{{\left\| \frac{pa+qb}{2} \right\| }^{r}} \right) , \end{aligned} \end{aligned}

where $$r\ge 1$$, $$\lambda =\min \left\{ {1}/{p},{1}/{q}\right\}$$ and $$\mu =\max \left\{ {1}/{p},{1}/{q} \right\}$$. Then we explain how this result extends some known results in the literature. Many other related results will be also shown. Then, with the theme of convexity, we employ a log-convex approach on certain matrix functions to obtain improvements and new sights of some matrix inequalities, including possible bounds of $$\Vert A^{t}XB^{1-t}\Vert ,$$ where AB are positive definite matrices, X is an arbitrary matrix, $$\Vert \cdot \Vert$$ is a unitarily invariant norm and $$0\le t\le 1.$$ Many other results involving matrix and scalar log-convex functions will be presented too.

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## References

1. Aldaz, J.M.: Strengthened Cauchy–Schwarz and Hölder inequalities, J. Inequal. Pure Appl. Math. 10(4), Article 116 (2009)

2. Al-Natoor, A., Audeh, W.: Refinement of triangle inequality for the Schatten $$p$$-norm. Adv. Oper. Theory 5, 1635–1645 (2020)

3. Bhatia, R., Davis, C.: A Cauchy–inequality for operators with applications. Linear Algebra Appl. 223, 119–129 (1995)

4. Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40, 396–414 (1936)

5. Dadipour, F., Moslehian, M.S., Rassias, J.M., Takahasi, S.E.: Characterization of a generalized triangle inequality in normed spaces. Nonlinear Anal. 75, 735–741 (2012)

6. Davarpanah, S.M., Moradi, H.R.: A log-convex approach to Jensen–Mercer inequality. J. Linear. Topol. Algebra 11(3), 169–176 (2022)

7. Dehghan, H.: A characterization of inner product spaces related to the skew-angular distance. Math. Notes 93(4), 556–560 (2013)

8. Diminnie, C.R., Andalafte, E.Z., Freese, R.W.: Angles in normed linear spaces and a characterization of real inner product spaces. Math. Nachr. 129, 197–204 (1986)

9. Dragomir S.S.: A potpourri of Schwarz related inequalities in inner product spaces (II), J. Inequal. Pure Appl. Math. 7(1), Article 14 (2006)

10. Dragomir, S.S.: Bounds for the normalised Jensen functional. Bull. Austral. Math. Soc. 74, 471–478 (2006)

11. Dragomir, S.S.: Upper and lower bounds for the $$p$$-angular distance in normed spaces with applications. J. Math. Inequal. 8, 947–961 (2014)

12. Dunkl, C.F., Wiliams, K.S.: A simple norm inequality. Am. Math. Monthly 71, 53–54 (1964)

13. Fujii, M., Nakamoto, R.: Refinements of Holder–McCarthy inequality and Young inequality. Adv. Oper. Theory 1(2), 184–188 (2016)

14. Krnić, M., Minculete, N.: Bounds for the $$p$$-angular distance and characterizations of inner product spaces. Mediterr. J. Math. 18, 140 (2021)

15. Krnić, M., Minculete, N.: Characterizations of inner product spaces via angular distances and Cauchy-Schwarz inequality. Aequat. Math. 95, 147–166 (2021)

16. Maligranda, L.: Simple norm inequalities. Amer. Math. Monthly 113, 256–260 (2006)

17. Maligranda, L.: Some remarks on the triangle inequality for norms. Banach J. Math. Anal. 2, 31–41 (2008)

18. Massera, J.L., Schäffer, J.J.: Linear differential equations and functional analysis, I. Ann. Math. 67(3), 517–573 (1958)

19. Minculete, N., Moradi, H.R.: Some improvements of the Cauchy-Schwarz inequality using the Tapia semi-inner-product. Mathematics. 8, 2112 (2020)

20. Minculete, N., Păltănea, R.: Improved estimates for the triangle inequality. J. Inequal. Appl. 2017, 17 (2017)

21. Moradi, H.R., Sababheh, M.: More accurate numerical radius inequalities (II). Linear Multilinear Algebra. 69(5), 921–933 (2021)

22. Moradi, H.R., Furuichi, S., Sababheh, M.: Some operator inequalities via convexity. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2021.2006592.

23. Moslehian, M.S., Dadipour, F., Rajić, R., Marić, A.: A glimpse at the Dunkl-Williams inequality. Banach J. Math. Anal. 5, 138–151 (2011)

24. Pečarić, J., Rajić, R.: On some generalized norm triangle inequalities. Rad HAZU. 515, 43–52 (2013)

25. Rooin, J., Rajabi, S., Moslehian, M.S.: $$p$$-angular distance orthogonality. Aequationes Math. 94(1), 103–121 (2020)

26. Sababheh, M.: Log and Harmonically log-convex functions related to matrix norms. Oper. Matrix 10(2), 453–465 (2016)

27. Sababheh, M.: Means refinements via convexity. Mediterr. J. Math. 14, 125 (2017). https://doi.org/10.1007/s00009-017-0924-8

28. Scharnhorst, K.: Angles in complex vector spaces. Acta Appl. Math. 69, 95–103 (2001)

29. Wigren, T.: The Cauchy-Schwarz inequality: Proofs and applications in various spaces, monograph

## Acknowledgements

The authors would like to express their deep gratitude to the reviewer, for valuable comments.

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Conde, C., Minculete, N., Moradi, H.R. et al. Norm Inequalities via Convex and Log-Convex Functions. Mediterr. J. Math. 20, 6 (2023). https://doi.org/10.1007/s00009-022-02214-z

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• DOI: https://doi.org/10.1007/s00009-022-02214-z