AIMS Mathematics, 2020, 5(6): 6108-6123. doi: 10.3934/math.2020392.

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Some unified bounds for exponentially $tgs$-convex functions governed by conformable fractional operators

1 School of Science, Huzhou University, Huzhou 313000, P. R. China
2 Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
3 Department of Mathematics, COMSATS University, Islamabad 44000, Pakistan
4 Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
5 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

In the article, we introduce the concept of the exponentially $tgs$-convex function and discover two new conformable fractional integral identities concerning the first-order differentiable convex mappings. By using these identities, we establish several new right-sided Hermite-Hadamard type inequalities for the exponentially $tgs$-convex functions via conformable fractional integrals. Our outcomes for conformable fractional integral operators are also applied to some special means.
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Keywords integral inequality; exponentially $tgs$-convex function; conformable fractional integral operator; Hermite-Hadamard inequality

Citation: Hu Ge-JiLe, Saima Rashid, Muhammad Aslam Noor, Arshiya Suhail, Yu-Ming Chu. Some unified bounds for exponentially $tgs$-convex functions governed by conformable fractional operators. AIMS Mathematics, 2020, 5(6): 6108-6123. doi: 10.3934/math.2020392


  • 1. S. Kumar, R. Kumar, C. Cattani, et al. Chaotic behaviour of fractional predator-prey dynamical system, Chaos Solutons Fractals, 135 (2020), 1-12.
  • 2. M. A. Akinlar, F. Tchier, M. Inc, Chaos control and solutions of fractional-order Malkus waterwheel model, Chaos Solitons Fractals, 135 (2020), 1-8.
  • 3. Y. Khurshid, M. Adil Khan, Y. M. Chu, Conformable fractional integral inequalities for GG- and GA-convex function, AIMS Math., 5 (2020), 5012-5030.    
  • 4. S. Rafeeq, H. Kalsoom, S. Hussain, et al. Delay dynamic double integral inequalities on time scales with applications, Adv. Differ. Equ., 2020 (2020), 1-32.    
  • 5. S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Math., 5 (2020), 2629-2645.    
  • 6. S. Rashid, İ. İşcan, D. Baleanu, et al. Generation of new fractional inequalities via n polynomials s-type convexixity with applications, Adv. Differ. Equ., 2020 (2020), 1-20.    
  • 7. S. S. Zhou, S. Rashid, F. Jarad, et al. New estimates considering the generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 2020 (2020), 1-15.    
  • 8. A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.
  • 9. S. Rashid, F. Jarad, M. A. Noor, et al. Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2019), 1-18.
  • 10. S. Rashid, F. Jarad, Y. M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Probl. Eng., 2020 (2020), 1-12.
  • 11. S. Rashid, F. Jarad, H. Kalsoom, et al. On Pólya-Szegö and Ćebyšev type inequalities via generalized k-fractional integrals, Adv. Differ. Equ., 2020 (2020), 1-18.    
  • 12. M. U. Awan, S. Talib, Y. M. Chu, et al. Some new refinements of Hermite-Hadamard-type inequalities involving Ψk-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 1-10.
  • 13. D. Baleanu, M. Jleli, S. Kumar, et al. A fractional derivative with two singular kernels and application to a heat conduction problem, Advs. Differ. Equ., 2020 (2020), 1-19.    
  • 14. J. Singh, D. Kumar, S. Kumar, An efficient computational method for local fractional transport equation occurring in fractal porous media, Comput. Appl. Math., 39 ((2020), 1-10.
  • 15. S. Kumar, A. Kumar, Z. Odibat, et al. A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow, AIMS Math., 5 (2020), 3035-3055.    
  • 16. R. Kumar, S. Kumar, J. Singh, et al. A comparative study for fractional chemical kinetics and carbon dioxide Co2 absorbed into phenyl glycidyl ether problems, AIMS Math., 5 (2020), 3201-3222.    
  • 17. M. Inc, A. Yusuf, A. I. Aliyu, et al. Dark and singular optical solitons for the conformable space-time nonlinear Schrödinger equation with Kerr and power law onlinearity, Optik, 162 (2018), 65-75.    
  • 18. Z. Korpinar, M. Inc, Numerical simulations for fractional variation of (1 + 1)-dimensional Biswas-Milovic equation, Optik, 166(218), 77-85.
  • 19. P. Agarwal, M. Kadakal, İ. İşcan, et al. Better approaches for n-times differentiable convex functions, Mathematics, 8 (2020), 1-11.
  • 20. T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), 1-14.    
  • 21. I. Abbas Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Space., 2020 (2020), 1-7.
  • 22. M. K. Wang, Z. Y. He, Y. M. Chu, Sharp power mean inequalities for the generalized elliptic integral of the first kind, Comput. Meth. Funct. Th., 20 (2020), 111-124.    
  • 23. S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Math., 5 (2020), 3525-3546.    
  • 24. M. Adil Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14.    
  • 25. T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math., 5 (2020), 4512-4528.    
  • 26. M. U. Awan, N. Akhtar, A. Kashuri, et. al. 2D approximately reciprocal ρ-convex functions and associated integral inequalities, AIMS Math., 5 (2020), 4662-4680.    
  • 27. S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10.    
  • 28. R. Khalil, M. A. Horani, A. Yousaf, et al. New definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.    
  • 29. S. Khan, M. Adil Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Method. Appl. Sci., 43 (2020), 2577-2587.    
  • 30. M. Adil Khan, J. Pečarić, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4931-4945.    
  • 31. T. H. Zhao, Y. M. Chu, H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011 (2011), 1-13.
  • 32. Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.
  • 33. M. K. Wang, M. Y. Hong, Y. F. Xu, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1-21.
  • 34. M. K. Wang, H. H. Chu, Y. M. Li, et al. Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind, Appl. Anal. Discrete Math., 14 (2020), 255-271.
  • 35. M. K. Wang, Y. M. Chu, Y. P. Jiang, Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions, Rocky Mt. J. Math., 46 (2016), 679-691.    
  • 36. M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 1-9.
  • 37. W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 1-12.    
  • 38. M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33.    
  • 39. M. U. Awan, N. Akhtar, S. Iftikhar, et al. New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12.    
  • 40. M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20.    
  • 41. S. Rashid, M. A. Noor, K. I. Noor, et al. Hermite-Hadamrad type inequalities for the class of convex functions on time scale, Mathematics, 7 (2019), 1-20.
  • 42. S. Bernstein, Sur les fonctions absolument monotones, Acta Math., 52 (1929), 1-66.    
  • 43. M. Avriel, r-convex functions, Math. Programming, 2 (1972), 309-323.    
  • 44. J. Jakšetić, J. Pečarić, Exponential convexity method, J. Convex Anal., 20 (2013), 181-197.
  • 45. T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.    
  • 46. F. Jarad, E. Uǧurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1-16.    
  • 47. M. U. Awan, M. A. Noor, K. I. Noor, Hermite-Hadamard inequalitie for exponentially convex function, Appl. Math. Inf. Sci., 12 (2018), 405-409.    
  • 48. Y. M. Chu, Y. F. Qiu, M. K. Wang, Hölder mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 23 (2012), 521-527.    
  • 49. G. D. Wang, X. H. Zhang, Y. M. Chu, A power mean inequality for the Grötzsch ring function, Math. Inequal. Appl., 14 (2011), 833-837.
  • 50. M. K. Wang, Y. M. Chu, Y. F. Qiu, et al. An optimal power mean inequality for the complete elliptic integrals, Appl. Math. Lett., 24 (2011), 887-890.    
  • 51. H. Z. Xu, Y. M. Chu, W. M. Qian, Sharp bounds for the Sándor-Yang means in terms of arithmetic and contra-harmonic means, J. Inequal. Appl., 2018 (2018), 1-13.    
  • 52. B. Wang, C. L. Luo, S. H. Li, et al. Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, RACSAM, 114 (2020), 1-10.    
  • 53. W. M. Qian, W. Zhang, Y. M. Chu, Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes, 20 (2019), 1157-1166.    
  • 54. W. M. Qian, Y. Y. Yang, H. W. Zhang, et al. Optimal two-parameter geometric and arithmetic mean bounds for the Sándor-Yang mean, J. Inequal. Appl., 2019 (2019), 1-12.    
  • 55. W. M. Qian, Z. Y. He, H. W. Zhang, et al. Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean, J. Inequal. Appl., 2019 (2019), 1-13.    
  • 56. W. M. Qian, X. H. Zhang, Y. M. Chu, Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 11 (2017), 121-127.
  • 57. Y. M. Chu, M. K. Wang, Optimal Lehmer mean bounds for the Toader mean, Results Math., 61 (2012), 223-229.    
  • 58. Y. M. Chu, M. K. Wang, S. L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 122 (2012), 41-51.    


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