Some Novel Integral Inequalities on the Co-ordinates for Geometrically Exponentially Convex Functions

Öz

The main purpose of this study is to define geometrically exponentially convex functions, which are a more general version, by expanding geometrically convex functions and to create the relevant lemmas. Some properties of geometrically exponentially convex functions are proven using definitions and lemmas. While obtaining the main findings, in addition to basic analysis information, Young and Hölder inequalities, well known in the literature, were also used for the powers of some functions. In the new theorems obtained, some special results were obtained for α=0.

Anahtar Kelimeler

• Hermite-Hadamard
• Geometrically exponentially convex functions
• Inequalities in the Coordinates

Kaynakça

1. Awan MU, Noor MA, Noor KI. Hermite-Hadamard Inequalities for Exponentially Convex Functions, Appl. Math. Inf. Sci. 2018; 12, No. 2, 405-409.
2. Peajcariaac JE, Tong YL. Convex functions, partial orderings, and statistical applications. Academic Press;1992.
3. Niculescu CP. Convexity according to the geometric mean. Math. Inequal. Appl. 2000; 3(2): 155-167.
4. Özdemir ME, Yıldız Ç, Gürbüz M. A note on geometrically convex functions. Journal of Inequalities and Applications. 2014(1): 1-12.
5. Xi BY, Bai RF, Qi F. Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions. Aequationes mathematicae, 2012; 84(3): 261-269.
6. Özdemir ME. Inequalities on geometrically convex functions. arXiv preprint arXiv: 2013; 1312.7725.
7. Dokuyucu M, Aslan S. Some New Approaches for Geometrically Convex Functions. 9ROXPH. 2022; 156.
8. Rashid S, Akdemir AO, Ekinci A, Aslan S. Some Fractional Integral Inequalities for Geometrically Convex Functions. In 3rd International Conference on Mathematical and Related Sciences: Current Trends and Developments Proceedings Book. (2020, November). (p. 246).

9. Butt SI, Ekinci A, Akdemir AO, Aslan S. Inequalities for Geometrically Convex Functions. In 3rd International Conference on Mathematical and Related Sciences: Current Trends and Developments Proceedings Book. (2020, November). (Vol. 1, p. 238).
10. Akdemir AO, Dutta H. (2020). New integral inequalities for product of geometrically convex functions. In 4th International Conference on Computational Mathematics and Engineering Sciences (CMES-2019) 4 (pp. 315-323). Springer International Publishing.
11. Aslan S, Akdemir AO. (2022, August). Exponentially convex functions on the co-ordinates and related integral inequalities. In Proceedings of the 8th International Conference on Control and Optimization with Industrial Applications (Vol. 2, pp. 120-122).
12. Dragomir SS. On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese Journal of Math., 5, 2001, 775-788.
13. Aslan S. (2023). Eksponansiyel konveks fonksiyonlar için koordinatlarda integral eşitsizlikler. PhD thesis, Doktora Tezi, Ağri İbrahim Çeçen Üniversitesi, Lisansüstü Eğitim Enstitüsü.
14. Akdemir AO, Aslan S, Dokuyucu MA, Çelik E. (2023). Exponentially Convex Functions on the Coordinates and Novel Estimations via Riemann-Liouville Fractional Operator. Journal of Function Spaces, 2023.
15. Aslan S, Akdemir AO. (2023). Exponential s-Convex Functions in the First Sense on the Co-ordinates and Some Novel Integral Inequalities. Turkish Journal of Science, 8(2), 85-92.
16. Aslan S, Akdemir AO, Dokuyucu MA. (2022). Exponentiallym- and (α,m)- Convex Functions on the Coordinates and Related Inequalities. Turkish Journal of Science, 7(3), 231-244.
17. Akdemir AO, Aslan S, Set E. (2022, October). Some New Inequalities for Exponentially Quasi-Convex Functions on the Coordinates and Related Hadamard Type Integral Inequalities. In 5th International Conference on Mathematical and Related Sciences Book of Proceedings (p. 120).
18. Akdemir AO, Aslan S, Ekinci A. (2022, October). Some New Inequalities for Exponentially P-Functions on the Coordinates. In 5th International Conference on Mathematical and Related Sciences Book of Proceedings (94-108).
19. Nie D, Rashid S, Akdemir AO, Baleanu D, Liu JB. On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications. Mathematics. 2019;7(8), 727.
20. Rashid S, Noor MA, Noor KI, Akdemir AO. Some new generalizations for exponentially s-convex functions and inequalities via fractional operators. Fractal and fractional. 2019;3(2): 24.
21. Rashid S, Safdar F, Akdemir AO, Noor MA, Noor KI. Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function. Journal of Inequalities and Applications. 2019; 1-17.
22. Anderson GD, Vamanamurthy MK, Vuorinen M. Generalized convexity and inequalities, J. Math. Anal. Appl., 335 (2007), 1294-1308.
23. Bullen PS. Handbook of Means and Their Inequalities, Math. Appl., vol. 560, Kluwer Academic Publishers, Dordrecht, 2003.
24. Bullen PS, Mitrinović DS, Vasic PM. Means and Their Inequalities, Reidel, Dordrecht; 1988.