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The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function

Year 2025, Volume: 54 Issue: 2, 404 - 413, 28.04.2025

Gültekin Tınaztepe , Sinem Sezer Evcan , Zeynep Eken , Sevda Sezer

https://doi.org/10.15672/hujms.1444589

Abstract

By means of an integral identity, several Hermite-Hadamard type inequalities are presented in this study for a function whose derivative's absolute value is the log-p-convex function. With the use of these findings, we are able to determine the boundaries in terms of elementary functions for certain specific functions, such as the imaginary error function, the exponential integral, the hyperbolic sine and cosine functions. Additionally, a relationship between beta function, the hyperbolic sine and cosine functions is stated. Through the obtained results, a bound for numerical integration of such type functions is provided.

Keywords

p-convex set log p-convex function p-convex function Hermite-Hadamard inequality

References

  • [1] G. Adilov and I. Yesilce, $B^{-1}$-Convex Functions, J. Convex Anal. 24 (2), 505-517, 2017.
  • [2] H. Boche and M. Schubert, A Calculus for log-Convex Interference Functions, IEEE Transactions on Information Theory 54 (12), 5469-5490, 2008.
  • [3] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/frvsxdcdwg > [Accessed 9 April 2023]
  • [4] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/tpi5wutstg > [Accessed 9 April 2023]
  • [5] S. S. Dragomir and C. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Math. Prep. Archive, 3, 463-817, 2003.
  • [6] Z. Eken, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Fourth Sense and Some of Their Properties, Konuralp J. Math. 9 (2), 260-267, 2021.
  • [7] S. Kemali, Hermite-Hadamard Type Inequality For s-Convex Functions in the Fourth Sense, Turkish J. Math. Comp. Sci. 13 (2), 287-293, 2021.
  • [8] S. Kemali, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Third Sense. Korean J. Math. 29 (3) , 593-602, 2021.
  • [9] S. Kemali, S. Sezer Evcan, I. Yesilce Isik and G. Adilov, Some Integral Inequalities for the Product of s-Convex Functions in the Fourth Sense. Int. J. Nonlinear Anal. Appl. 13 (2), 103-116, 2022.
  • [10] S. Kemali, I. Yesilce and G. Adilov, $\mathbb{B}$ -Convexity, $\mathbb{B}^{-1}$-Convexity, And Their Comparison, Numer. Funct. Anal. Optim. 36 (2), 133-146, 2015.
  • [11] A. Klinger and O. L. Mangasarian, Logarithmic Convexity and Geometric Programming, J. Mat. Anal. Appl. 24 (2), 388-408, 1968.
  • [12] R. J. Knops, Logarithmic Convexity and Other Techniques Applied to Problems in Continuum Mechanics. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity: Held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972 (31-54), Springer Berlin Heidelberg, 2006.
  • [13] H. Mamedov and I. Yesilce Isik, On the Fractional Integral Inequalities for p-convex Functions, Punjab Univ. J. Math. 55 (5-6), 185-196, 2023.
  • [14] M. A. Noor and K. I. Noor, New Perspectives of log-Convex Functions. Applied Math. Inf. Sci. 14 (5), 847-854, 2020.
  • [15] M. A. Noor and K. I. Noor, Strongly log-Convex Functions. Information Sciences Letters 10 (1), 33-38, 2021.
  • [16] R. Quintanilla, On the Logarithmic Convexity in Thermoelasticity with Microtemperatures. Journal of Thermal Stresses. 36 (4), 378-386, 2013.
  • [17] C. P. Rydell, The Significance of Logarithmic Convexity for Price and Growth Theory, Western Economic Journal Oxford, 6 (1), 65-71, 1967.
  • [18] S. Sezer, Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are p-Convex. Fundamental Journal of Mathematics and Applications, 4 (2), 88-99, 2021.
  • [19] S. Sezer, Z. Eken, G. Tınaztepe and G. Adilov, p-Convex Functions and Their Some Properties, Numer. Funct. Anal. Optim. 42 (4), 443-459, 2021.
  • [20] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov. Quasi p-Convex Functions. Appl. Math. E-Notes, 22, 741-750, 2022.
  • [21] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov, Logarithmic p-Convex Functions and Some of Their Properties, 2024, (submitted).
  • [22] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities For L (J)-Convex Functions And S (J)-Convex Functions, Malaya Journal of Matematik, 3 (3), 346-359, 2015.
  • [23] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities for $\mathbb{B}$-Convex and $\mathbb{B}^{-1}$- Convex Functions. International Journal of Nonlinear Analysis and Applications, 8 (1), 225-233, 2017.

Year 2025, Volume: 54 Issue: 2, 404 - 413, 28.04.2025

Gültekin Tınaztepe , Sinem Sezer Evcan , Zeynep Eken , Sevda Sezer

https://doi.org/10.15672/hujms.1444589

Abstract

References

  • [1] G. Adilov and I. Yesilce, $B^{-1}$-Convex Functions, J. Convex Anal. 24 (2), 505-517, 2017.
  • [2] H. Boche and M. Schubert, A Calculus for log-Convex Interference Functions, IEEE Transactions on Information Theory 54 (12), 5469-5490, 2008.
  • [3] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/frvsxdcdwg > [Accessed 9 April 2023]
  • [4] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/tpi5wutstg > [Accessed 9 April 2023]
  • [5] S. S. Dragomir and C. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Math. Prep. Archive, 3, 463-817, 2003.
  • [6] Z. Eken, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Fourth Sense and Some of Their Properties, Konuralp J. Math. 9 (2), 260-267, 2021.
  • [7] S. Kemali, Hermite-Hadamard Type Inequality For s-Convex Functions in the Fourth Sense, Turkish J. Math. Comp. Sci. 13 (2), 287-293, 2021.
  • [8] S. Kemali, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Third Sense. Korean J. Math. 29 (3) , 593-602, 2021.
  • [9] S. Kemali, S. Sezer Evcan, I. Yesilce Isik and G. Adilov, Some Integral Inequalities for the Product of s-Convex Functions in the Fourth Sense. Int. J. Nonlinear Anal. Appl. 13 (2), 103-116, 2022.
  • [10] S. Kemali, I. Yesilce and G. Adilov, $\mathbb{B}$ -Convexity, $\mathbb{B}^{-1}$-Convexity, And Their Comparison, Numer. Funct. Anal. Optim. 36 (2), 133-146, 2015.
  • [11] A. Klinger and O. L. Mangasarian, Logarithmic Convexity and Geometric Programming, J. Mat. Anal. Appl. 24 (2), 388-408, 1968.
  • [12] R. J. Knops, Logarithmic Convexity and Other Techniques Applied to Problems in Continuum Mechanics. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity: Held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972 (31-54), Springer Berlin Heidelberg, 2006.
  • [13] H. Mamedov and I. Yesilce Isik, On the Fractional Integral Inequalities for p-convex Functions, Punjab Univ. J. Math. 55 (5-6), 185-196, 2023.
  • [14] M. A. Noor and K. I. Noor, New Perspectives of log-Convex Functions. Applied Math. Inf. Sci. 14 (5), 847-854, 2020.
  • [15] M. A. Noor and K. I. Noor, Strongly log-Convex Functions. Information Sciences Letters 10 (1), 33-38, 2021.
  • [16] R. Quintanilla, On the Logarithmic Convexity in Thermoelasticity with Microtemperatures. Journal of Thermal Stresses. 36 (4), 378-386, 2013.
  • [17] C. P. Rydell, The Significance of Logarithmic Convexity for Price and Growth Theory, Western Economic Journal Oxford, 6 (1), 65-71, 1967.
  • [18] S. Sezer, Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are p-Convex. Fundamental Journal of Mathematics and Applications, 4 (2), 88-99, 2021.
  • [19] S. Sezer, Z. Eken, G. Tınaztepe and G. Adilov, p-Convex Functions and Their Some Properties, Numer. Funct. Anal. Optim. 42 (4), 443-459, 2021.
  • [20] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov. Quasi p-Convex Functions. Appl. Math. E-Notes, 22, 741-750, 2022.
  • [21] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov, Logarithmic p-Convex Functions and Some of Their Properties, 2024, (submitted).
  • [22] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities For L (J)-Convex Functions And S (J)-Convex Functions, Malaya Journal of Matematik, 3 (3), 346-359, 2015.
  • [23] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities for $\mathbb{B}$-Convex and $\mathbb{B}^{-1}$- Convex Functions. International Journal of Nonlinear Analysis and Applications, 8 (1), 225-233, 2017.
There are 23 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Mathematics
Authors

Gültekin Tınaztepe 0000-0001-7594-1620

Sinem Sezer Evcan 0000-0003-2066-7833

Zeynep Eken 0000-0002-8939-4653

Sevda Sezer 0000-0001-6448-193X

Early Pub Date August 27, 2024
Publication Date April 28, 2025
Submission Date February 28, 2024
Acceptance Date April 23, 2024
Published in Issue Year 2025 Volume: 54 Issue: 2

Cite

APA Tınaztepe, G., Sezer Evcan, S., Eken, Z., Sezer, S. (2025). The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics, 54(2), 404-413. https://doi.org/10.15672/hujms.1444589
AMA Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. April 2025;54(2):404-413. doi:10.15672/hujms.1444589
Chicago Tınaztepe, Gültekin, Sinem Sezer Evcan, Zeynep Eken, and Sevda Sezer. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics 54, no. 2 (April 2025): 404-13. https://doi.org/10.15672/hujms.1444589.
EndNote Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S (April 1, 2025) The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics 54 2 404–413.
IEEE G. Tınaztepe, S. Sezer Evcan, Z. Eken, and S. Sezer, “The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, pp. 404–413, 2025, doi: 10.15672/hujms.1444589.
ISNAD Tınaztepe, Gültekin et al. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics 54/2 (April 2025), 404-413. https://doi.org/10.15672/hujms.1444589.
JAMA Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. 2025;54:404–413.
MLA Tınaztepe, Gültekin et al. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, 2025, pp. 404-13, doi:10.15672/hujms.1444589.
Vancouver Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):404-13.

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