In this paper, the Hermite-Hadamard inequality and its generalization for quasi $ p $-convex functions are provided. Also several new inequalities are established for the functions whose first derivative in absolute value is quasi $ p $-convex, which states some bounds for sides of the Hermite-Hadamard inequalities. In the context of the applications of results, we presented some relations involving special means and some inequalities for special functions including digamma function and Fresnel integral for sinus. In addiditon, an upper bound for error in numerical integration of quasi p-convex functions via composite trapezoid rule is given.
Citation: Sevda Sezer, Zeynep Eken. The Hermite-Hadamard type inequalities for quasi $ p $-convex functions[J]. AIMS Mathematics, 2023, 8(5): 10435-10452. doi: 10.3934/math.2023529
In this paper, the Hermite-Hadamard inequality and its generalization for quasi $ p $-convex functions are provided. Also several new inequalities are established for the functions whose first derivative in absolute value is quasi $ p $-convex, which states some bounds for sides of the Hermite-Hadamard inequalities. In the context of the applications of results, we presented some relations involving special means and some inequalities for special functions including digamma function and Fresnel integral for sinus. In addiditon, an upper bound for error in numerical integration of quasi p-convex functions via composite trapezoid rule is given.
[1] | G. R. Adilov, S. Kemali, Abstract convexity and Hermite-Hadamard type inequalities, J. Inequal. Appl., 2009 (2009), 943534. https://doi.org/10.1155/2009/943534 doi: 10.1155/2009/943534 |
[2] | G. Adilov, I. Yesilce, $B^{-1}$-convex sets and $B^{-1}$-measurable maps, Numer. Funct. Anal. Opt., 33 (2012), 131–141. https://doi.org/10.1016/j.ppedcard.2012.02.006 doi: 10.1016/j.ppedcard.2012.02.006 |
[3] | G. Adilov, A. M. Rubinov, $B$-convex sets and functions, Numer. Funct. Anal. Opt., 27 (2006), 237–257. https://doi.org/10.1080/01630560600698343 doi: 10.1080/01630560600698343 |
[4] | P. S. Bullen, Handbook of means and their inequalities, Kluwer, Dordrecht, Netherlands, 2003. |
[5] | S. Chang, Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl., 159 (1991), 208–223. https://doi.org/10.1016/0022-247X(91)90231-N doi: 10.1016/0022-247X(91)90231-N |
[6] | S. S. Dragomir, C. E. Pearce, Quasi-convex functions and Hadamard's inequality, B. Aust. Math. Soc., 57 (1998), 377–385. https://doi.org/10.1017/S0004972700031786 doi: 10.1017/S0004972700031786 |
[7] | S. S. Dragomir, C. E. Pearce, Jensen's inequality for quasiconvex functions, Numer. Algebr. Control, 2 (2012), 279–291. https://doi.org/10.3934/naco.2012.2.279 doi: 10.3934/naco.2012.2.279 |
[8] | S. S. Dragomir, C. E. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Science Direct Working Paper, 04 (2003), S1574-0358. |
[9] | Z. Eken, S. Kemali, G. Tınaztepe, G. Adilov, The Hermite-Hadamard inequalities for $p$-convex functions, Hacet. J. Math. Stat., 50 (2021), 1268–1279. |
[10] | Z. Eken, S. Sezer, G. Tınaztepe, G. Adilov, s-Convex functions in the fourth sense and some of their properties, Konuralp J. Math., 9 (2021), 260–267. |
[11] | M. E. Gordji, M. R. Delavar, M. De La Sen, On $\phi$-convex functions, J. Math. Inequal., 10 (2016), 173–183. https://doi.org/10.7153/jmi-10-15 doi: 10.7153/jmi-10-15 |
[12] | M. Kadakal, Hermite-Hadamard type inequalities for quasi-convex functions via improved power-mean inequality, TWMS J. Appl. Eng. Math., 11 (2021), 194–202. https://doi.org/10.5458/bag.11.4_202 doi: 10.5458/bag.11.4_202 |
[13] | S. Kemali, S. Sezer, G. Tınaztepe, G. Adilov, $s$-Convex functions in the third sense, Korean J. Math., 29 (2021), 593–602. |
[14] | H. Lebedev, Special functions and their applications (Romanian), Ed.Tehhica. Bucharest, 1957. |
[15] | Z. Pavic, M. A. Ardic, The most important inequalities of $m$-convex functions, Turk. J. Math., 41 (2017), 625–635. https://doi.org/10.3906/mat-1604-45 doi: 10.3906/mat-1604-45 |
[16] | S. Sezer, The Hermite-Hadamard inequality for $s$-convex functions in the third sense, AIMS Math., 6 (2021), 7719–7732. https://doi.org/10.3934/math.2021448 doi: 10.3934/math.2021448 |
[17] | S. Sezer, Z. Eken, G. Tınaztepe, G. Adilov, $p$-convex functions and some of their properties, Numer. Funct. Anal. Opt., 42 (2021), 443–459. https://doi.org/10.1080/01630563.2021.1884876 doi: 10.1080/01630563.2021.1884876 |
[18] | S. Sezer, Hermite-Hadamard type inequalities for the functions whose absolute values of first derivatives are p-convex, Fund. J. Math. Appl., 4 (2021), 88–99. |
[19] | K. B. Stolarsky, The power and generalized logarithmic means, Am. Math. Mon., 8 (1980), 545–548. https://doi.org/10.1080/00029890.1980.11995086 doi: 10.1080/00029890.1980.11995086 |
[20] | J. Tabor, J. Tabor, M. Żołdak, On $\omega $-quasiconvex functions, Math. Inequal. Appl., 15 (2012), 845–857. https://doi.org/10.7153/mia-15-72 doi: 10.7153/mia-15-72 |
[21] | G. Tınaztepe, S. Sezer, Z. Eken, G. Adilov, Quasi $p$-convex functions, Appl. Math. E-Notes, 22 (2022), 741–750. |
[22] | I. Yesilce, G. Adilov, Hermite-Hadamard inequalities for $B$-convex and $B^{-1}$-convex functions, Int. J. Nonlinear Anal. Appl., 8 (2017), 225–233. |
[23] | I. Yesilce, G. Adilov, Hermite-Hadamard inequalities for $L_{j}$-convex functions and $S_{j}$-convex functions, Malaya J. Mat., 3 (2015), 346–359. |