On Generalizations of Hölder's and Minkowski's Inequalities

Uğur Selamet Kırmacı

doi:10.36753/mathenot.1150375

EN

On Generalizations of Hölder's and Minkowski's Inequalities

Abstract

We present the generalizations of Hölder's inequality and Minkowski's inequality along with the generalizations of Aczel's, Popoviciu's, Lyapunov's and Bellman's inequalities. Some applications for the metric spaces, normed spaces, Banach spaces, sequence spaces and integral inequalities are further specified. It is shown that $({\mathbb{R}}^n,d)$ and $\left(l_p,d_{m,p}\right)$ are complete metric spaces and $({\mathbb{R}}^n,{\left\|x\right\|}_m)$ and $\left(l_p,{\left\|x\right\|}_{m,p}\right)$ are $\frac{1}{m}-$Banach spaces. Also, it is deduced that $\left(b^{r,s}_{p,1},{\left\|x\right\|}_{r,s,m}\right)$ is a $\frac{1}{m}-$normed space.

Keywords

Aczel's inequality, Bellman's inequality, Hölder's inequality

References

[1] Beckenbach, E .F., Bellman, R.: Inequalities, Springer-Verlag, Berlin (1961).
[2] Royden, H. L.: Real analysis. Macmillan Publishing Co. Inc. New-York (1968).
[3] Yosida, K.: Functional analysis. Springer-Verlag Berlin, Heidelberg, New-York (1974).
[4] Bi¸sgin, M. C.: The binomial sequence spaces which include the spaces lp and l1 and geometric properties. J. Inequal. Appl.2016, 304 (2016).
[5] Ellidokuzo˘ glu, H. B., Demiriz, S., Köseo˘ glu, A.: On the paranormed binomial sequence spaces. Univers. J. Math. Appl. 1, 137-147 (2018).
[6] Niculescu, C. P., Persson, L-E.: Convex functions and their applications. Springer (2004).
[7] Agahi, H., Ouyang, Y., Mesiar, R., Pap, E., Štrboja, M.: Hölder and Minkowski type inequalities for pseudo-integral. Appl. Math. Comput. 217, 8630-8639 (2011).
[8] Zhao, C. J., Cheung,W. S.: On Minkowski’s inequality and its application. J. Inequal. Appl. 2011, 71 (2011).
[9] Zhou, X.: Some generalizations of Aczél, Bellman’s inequalities and related power sums. J. Inequal. Appl. 2012, 130 (2012).
[10] Butt, S. I., Horváth, L., Peˇcari´c, J.: Cyclic refinements of the discrete Hölder’s inequality with applications.Miskolc Math. Notes. 21, 679-687 (2020).

[11] Rashid, S., Hammouch, Z., Baleanu, D., Chu, Y. M.: New generalizations in the sense of the weighted non-singular fractional integral operatör. Fractals. 28, 2040003 (2020). https://doi.org/10.1142/S0218348X20400034.
[12] Rashid, S., Jarad, F., Chu, Y. M.: A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function. Math. Probl. Eng. 2020, 7630260 (2020). https://doi.org/10.1155/2020/7630260.
[13] Rashid, S., Sultana, S., Karaca, Y., Khalid, A., Chu, Y. M.: Some further extensions considering discrete proportional fractional operators. Fractals. 30, 2240026 (2022). https://doi.org/10.1142/S0218348X22400266.
[14] Rashid, S., Abouelmagd, E. I., Khalid, A., Farooq, F. B., Chu, Y. M.: Some recent developments on dynamical }}- discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels. Fractals. 30, 2240110 (2022). https://doi.org/10.1142/S0218348X22401107.
[15] Rafeeq, S., Kalsoom, H., Hussain, S., Rashid, S., Chu, Y. M.: Delay dynamic double integral inequalities on time scales with applications. Advances in Difference Equations. 2020, 40 (2020).
[16] Zong, Z., Hu, F., Yin, C., Wu, H.: On Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations. J. Inequal. Appl. 2015, 152 (2015). https://doi.org/10.1186/s13660- 015-0677-5.
[17] Alomari, M. W., Darus, M., Kirmaci, U. S.: Refinements of Hadamard type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means. Comput. Math. Appl. 59, 225-232 (2010).
[18] Alomari, M. W., Darus, M., Kirmaci, U. S.: Some inequalities of Hermite-Hadamard type for s-convex functions. Acta Math. Sci. Ser. B Eng. Ed. 31, 1643-1652 (2011).
[19] Bougoffa, L.: On Minkowski and Hardy integral inequalities. J. Inequal. Pure Appl. Math. 7, 60 (2006).
[20] Dragomir, S. S., Fitzpatrick, S.: s-Orlicz convex functions in linear spaces and Jensen’s discrete inequality. J. Math. Anal. Appl. 210, 419-439 (1997).
[21] Dragomir, S. S., Fitzpatrick, S.: The Hadamard inequalities for s-convex functions in the second sense. Demonstr. Math. 32, 687-696 (1999).
[22] Dragomir, S. S., Pearce, C. E. M.: Selected topics on Hermite-Hadamard inequalities and applications. RGMIA Monographs, Victoria University (2000). [online], http://www.staff.vu.edu.au/RGMIA/monographs/hermite hadamard.html.
[23] Dragomir, S. S.: Refining Hölder integral inequality for partitions of weights. RGMIA Res. Rep. Coll. 23, 1 (2020).
[24] Frenkel, P. E., Horváth, P.: Minkowski’s inequality and sums of squares. Cent. Eur. J. Math. 12, 510-516 (2014). https://doi.org/10.2478/s11533-013-0346-1.
[25] Hinrichs, A., Kolleck, A., Vybiral, J.: Carl’s inequality for quasi-Banach spaces. J. Funct. Anal. 271, 2293-2307 (2016).
[26] Kadakal M.: (m1;m2)-Geometric arithmetically convex functions and related inequalities. Math. Sci. Appl. E-Notes. 10, 63-71 (2022). https://doi.org/10.36753/mathenot.685624.
[27] Kemper, R.: p-Banach spaces and p-totally convex spaces. Applied Categorical Structures. 7, 279-295 (1999).
[28] Kirmaci, U. S., Bakula, M. K., Özdemir, M. E., Peˇcari´c J. E.: On some inequalities for p- norms. J. Inequal. Pure Appl. Math. 9, 27 (2008).
[29] Kirmaci, U. S., Bakula, M. K., Özdemir, M. E., Pe˘cari´c, J. E.: Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193, 26-35 (2007).
[30] Kirmaci, U. S.: Improvement and further generalization of inequalities for differentiable mappings and applications. Comput. Math. Appl. 55, 485-493 (2008).
[31] Kirmaci, U. S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137-146 (2004).
[32] Kirmaci, U. S., Özdemir, M. E.: On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 153, 361-368 (2004).
[33] Kirmaci, U. S., Özdemir, M. E.: Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers. Appl. Math. Lett. 17, 641-645 (2004).
[34] Kirmaci, U. S.: Refinements of Hermite-Hadamard type inequalities for s-convex functions with applications to special means. Univers. J. Math. Appl. 4, 114-124 (2021). https://doi.org/10.32323/ujma.953684.
[35] Ma, X. F., Wang, L. C.: Two mappings related to Minsowski’s inequality. J. Inequal. Pure Appl. Math. 10, 89 (2009).
[36] Mitrinovi´c, D. S.: Analytic inequalities. Springer-Verlag Berlin, Heidelberg, New-York (1970).
[37] Mitrinovi´c, D. S., Pe˘cari´c, J. E., Fink, A. M.: Classical and new inequalities in analysis. Kluwer Academic Publishers, London (1993).
[38] Sigg, M.: A Minkowski-type inequality for the Schatten norm. J. Inequal. Pure Appl. Math. 6, 87, (2005).
[39] Tunç, M., Kirmaci, U. S.: New integral inequalities for s-convex functions with applications. Int. Electron. J. Pure Appl. Math. 1, 131-141 (2010).
[40] Yang, X.: A note on Hölder inequality. Appl. Math. Comput. 134, 319-322 (2003).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Uğur Selamet Kırmacı ^*
0000-0002-8177-6649
Türkiye

Early Pub Date

August 8, 2023

Publication Date

October 25, 2023

Submission Date

July 29, 2022

Acceptance Date

January 24, 2023

Published in Issue

Year 2023 Volume: 11 Number: 4

DOI

https://doi.org/10.36753/mathenot.1150375

IZ

https://izlik.org/JA24PA46GR

APA

Kırmacı, U. S. (2023). On Generalizations of Hölder’s and Minkowski’s Inequalities. Mathematical Sciences and Applications E-Notes, 11(4), 213-225. https://doi.org/10.36753/mathenot.1150375

AMA

1.Kırmacı US. On Generalizations of Hölder’s and Minkowski’s Inequalities. Math. Sci. Appl. E-Notes. 2023;11(4):213-225. doi:10.36753/mathenot.1150375

Chicago

Kırmacı, Uğur Selamet. 2023. “On Generalizations of Hölder’s and Minkowski’s Inequalities”. Mathematical Sciences and Applications E-Notes 11 (4): 213-25. https://doi.org/10.36753/mathenot.1150375.

EndNote

Kırmacı US (October 1, 2023) On Generalizations of Hölder’s and Minkowski’s Inequalities. Mathematical Sciences and Applications E-Notes 11 4 213–225.

IEEE

[1]U. S. Kırmacı, “On Generalizations of Hölder’s and Minkowski’s Inequalities”, Math. Sci. Appl. E-Notes, vol. 11, no. 4, pp. 213–225, Oct. 2023, doi: 10.36753/mathenot.1150375.

ISNAD

Kırmacı, Uğur Selamet. “On Generalizations of Hölder’s and Minkowski’s Inequalities”. Mathematical Sciences and Applications E-Notes 11/4 (October 1, 2023): 213-225. https://doi.org/10.36753/mathenot.1150375.

JAMA

1.Kırmacı US. On Generalizations of Hölder’s and Minkowski’s Inequalities. Math. Sci. Appl. E-Notes. 2023;11:213–225.

MLA

Kırmacı, Uğur Selamet. “On Generalizations of Hölder’s and Minkowski’s Inequalities”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 4, Oct. 2023, pp. 213-25, doi:10.36753/mathenot.1150375.

Vancouver

1.Uğur Selamet Kırmacı. On Generalizations of Hölder’s and Minkowski’s Inequalities. Math. Sci. Appl. E-Notes. 2023 Oct. 1;11(4):213-25. doi:10.36753/mathenot.1150375