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On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity

Year 2020, Volume: 8 Issue: 1, 165 - 174, 15.04.2020

Sercan Turhan , Mehmet Kunt , İmdat İşcan

Abstract

In this paper, the authors give a new concept which is a generalization of the concepts $s$-convexity,$GA-s$-convexity, harmonically $s$-convexity and $(p,s)$-convexity establish some new Hermite-Hadamard type inequalities for this class of functions. Some natural applications to special means of real numbers are also given.

Keywords

Hermite-Hadamard type inequality

References

  • [1] J. Aczel , The notion of mean values, Norske Vid. Selsk. Forhdl., Trondhjem 19 (1947), 83–86.
  • [2] J. Aczel , A generalization of the notion of convex functions, Norske Vid. Selsk. Forhd., Trondhjem 19 (24) (1947), 87–90.
  • [3] G. Aumann , Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelverten, Bayer. Akad. Wiss.Math.-Natur. Kl. Abh., Math. Ann. 109 (1933), 405–413.
  • [4] M. Avcı,H. Kavurmacıand M. E. Ozdemir , New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., vol. 217 (2011), pp. 5171–5176.
  • [5] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen , Generalized convexity and inequalities, Journal of Mathematical Analysis and Applications 335 (2) (2007), 1294–1308.
  • [6] Y.-M. Chu , M. Adil Khan , T. U. Khan , and J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 15 (2017) 1414-1430.
  • [7] S.S. Dragomir , R.P. Agarwal , Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula, Appl. Math. Lett. 11 (5) (1998), 91–95.
  • [8] S. S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstr. Math., 32 (4) (1999), 687–696.
  • [9] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100–111.
  • [10] İ. İşcan , A new generalization of some integral inequalities for -convex functions, Mathematical Sciences 2013, 7:22,1–8.
  • [11] İ. İşcan, New estimates on generalization of some integral inequalities for s-convex functions and their applications, International Journal of Pure and Applied Mathematics, 86 (4) (2013), 727–746.
  • [12] İ. İşcan, Some New Hermite-Hadamard Type Inequalities for Geometrically Convex Functions, Mathematics and Statistics 1(2) (2013), 86–91.
  • [13] İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Journal of Mathematics and Statistics, 43 (6) (2014), 935–942.
  • [14] İ. İşcan, Some new general integral inequalities for h-convex and h-concave functions, Adv. Pure Appl. Math. 5 (1) (2014), 21–29.
  • [15] İ. İşcan, Hermite-Hadamard type inequalities for GA􀀀s-convex functions, Le Matematiche, Vol. LXIX (2014) – Fasc. II, pp. 129—146.
  • [16] İ. İşcan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, Studia Universitatis Babes¸-Bolyai Mathematica, 60(2015), no.3, 355–366.
  • [17] İ. İşcan, M. Kunt, Hermite-Hadamard-Fej´er type inequalities for harmonically s-convex functions via fractional integrals, The Australian Journal of Mathematical Analysis and Applications, Volume 12, Issue 1, Article 10, (2015), 1–16.
  • [18] İ. İşcan, Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences, NTMSCI 4 No. 3 (2016), 140–150.
  • [19] İ. İşcan, Hermite-Hadamard type inequalities for p-convex functions, International Journal of Analysis and Applications, Volume 11, Number 2 (2016), 137–145.
  • [20] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004), 137-146.
  • [21] A.A. Kilbas ,H.M. Srivastava and J.J. Trujillo , Theory and applications of fractional differential equations, Amsterdam, Elsevier, 2006.
  • [22] U. S. Kirmaci ,M. K. Bakula ,M. E. Ozdemir ,J. Pecaric, Hadamard-type inequalities for s-convex functions, Applied Mathematics and Computation 193 (2007) 26—35.
  • [23] M. Adil Khan ,T. Ali and T. U. Khan, Hermite-Hadamard Type Inequalities with Applications, Fasciculi Mathematici, 59 (2017), 57-74.
  • [24] Khan M. Adil Khan, T. Ali, M. Z. Sarikaya , and Q. Din, New bounds forHermite-Hadamard type inequalities with applications, Electronic Journal of Mathematical Analysis and Applications, to appear (2018).
  • [25] M. Adil Khan, Y. Khurshid , S. S. Dragomir and R. Ullah , Inequalities of the Hermite-Hadamard type with applications,Punjab Univ. J. Math., 50(3)(2018) 1-12.
  • [26] J. Matkowski , Convex functions with respect to a mean and a characterization of quasi-arithmetic means, Real Anal. Exchange 29 (2003/2004), 229–246.
  • [27] C. P. Niculescu , Convexity according to the geometric mean, Math. Inequal. Appl., vol. 3, no. 2 (2000), pp. 155–167.
  • [28] C.P. Niculescu, Convexity according to means, Math. Inequal. Appl. 6 (2003) 571–579.
  • [29] W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 9 (1961), 157-162.

Year 2020, Volume: 8 Issue: 1, 165 - 174, 15.04.2020

Sercan Turhan , Mehmet Kunt , İmdat İşcan

Abstract

References

  • [1] J. Aczel , The notion of mean values, Norske Vid. Selsk. Forhdl., Trondhjem 19 (1947), 83–86.
  • [2] J. Aczel , A generalization of the notion of convex functions, Norske Vid. Selsk. Forhd., Trondhjem 19 (24) (1947), 87–90.
  • [3] G. Aumann , Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelverten, Bayer. Akad. Wiss.Math.-Natur. Kl. Abh., Math. Ann. 109 (1933), 405–413.
  • [4] M. Avcı,H. Kavurmacıand M. E. Ozdemir , New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., vol. 217 (2011), pp. 5171–5176.
  • [5] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen , Generalized convexity and inequalities, Journal of Mathematical Analysis and Applications 335 (2) (2007), 1294–1308.
  • [6] Y.-M. Chu , M. Adil Khan , T. U. Khan , and J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 15 (2017) 1414-1430.
  • [7] S.S. Dragomir , R.P. Agarwal , Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula, Appl. Math. Lett. 11 (5) (1998), 91–95.
  • [8] S. S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstr. Math., 32 (4) (1999), 687–696.
  • [9] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100–111.
  • [10] İ. İşcan , A new generalization of some integral inequalities for -convex functions, Mathematical Sciences 2013, 7:22,1–8.
  • [11] İ. İşcan, New estimates on generalization of some integral inequalities for s-convex functions and their applications, International Journal of Pure and Applied Mathematics, 86 (4) (2013), 727–746.
  • [12] İ. İşcan, Some New Hermite-Hadamard Type Inequalities for Geometrically Convex Functions, Mathematics and Statistics 1(2) (2013), 86–91.
  • [13] İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Journal of Mathematics and Statistics, 43 (6) (2014), 935–942.
  • [14] İ. İşcan, Some new general integral inequalities for h-convex and h-concave functions, Adv. Pure Appl. Math. 5 (1) (2014), 21–29.
  • [15] İ. İşcan, Hermite-Hadamard type inequalities for GA􀀀s-convex functions, Le Matematiche, Vol. LXIX (2014) – Fasc. II, pp. 129—146.
  • [16] İ. İşcan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, Studia Universitatis Babes¸-Bolyai Mathematica, 60(2015), no.3, 355–366.
  • [17] İ. İşcan, M. Kunt, Hermite-Hadamard-Fej´er type inequalities for harmonically s-convex functions via fractional integrals, The Australian Journal of Mathematical Analysis and Applications, Volume 12, Issue 1, Article 10, (2015), 1–16.
  • [18] İ. İşcan, Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences, NTMSCI 4 No. 3 (2016), 140–150.
  • [19] İ. İşcan, Hermite-Hadamard type inequalities for p-convex functions, International Journal of Analysis and Applications, Volume 11, Number 2 (2016), 137–145.
  • [20] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004), 137-146.
  • [21] A.A. Kilbas ,H.M. Srivastava and J.J. Trujillo , Theory and applications of fractional differential equations, Amsterdam, Elsevier, 2006.
  • [22] U. S. Kirmaci ,M. K. Bakula ,M. E. Ozdemir ,J. Pecaric, Hadamard-type inequalities for s-convex functions, Applied Mathematics and Computation 193 (2007) 26—35.
  • [23] M. Adil Khan ,T. Ali and T. U. Khan, Hermite-Hadamard Type Inequalities with Applications, Fasciculi Mathematici, 59 (2017), 57-74.
  • [24] Khan M. Adil Khan, T. Ali, M. Z. Sarikaya , and Q. Din, New bounds forHermite-Hadamard type inequalities with applications, Electronic Journal of Mathematical Analysis and Applications, to appear (2018).
  • [25] M. Adil Khan, Y. Khurshid , S. S. Dragomir and R. Ullah , Inequalities of the Hermite-Hadamard type with applications,Punjab Univ. J. Math., 50(3)(2018) 1-12.
  • [26] J. Matkowski , Convex functions with respect to a mean and a characterization of quasi-arithmetic means, Real Anal. Exchange 29 (2003/2004), 229–246.
  • [27] C. P. Niculescu , Convexity according to the geometric mean, Math. Inequal. Appl., vol. 3, no. 2 (2000), pp. 155–167.
  • [28] C.P. Niculescu, Convexity according to means, Math. Inequal. Appl. 6 (2003) 571–579.
  • [29] W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 9 (1961), 157-162.
There are 29 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Sercan Turhan

Mehmet Kunt

İmdat İşcan

Publication Date April 15, 2020
Submission Date December 19, 2019
Acceptance Date March 28, 2020
Published in Issue Year 2020 Volume: 8 Issue: 1

Cite

APA Turhan, S., Kunt, M., & İşcan, İ. (2020). On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity. Konuralp Journal of Mathematics, 8(1), 165-174.
AMA Turhan S, Kunt M, İşcan İ. On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity. Konuralp J. Math. April 2020;8(1):165-174.
Chicago Turhan, Sercan, Mehmet Kunt, and İmdat İşcan. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics 8, no. 1 (April 2020): 165-74.
EndNote Turhan S, Kunt M, İşcan İ (April 1, 2020) On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity. Konuralp Journal of Mathematics 8 1 165–174.
IEEE S. Turhan, M. Kunt, and İ. İşcan, “On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity”, Konuralp J. Math., vol. 8, no. 1, pp. 165–174, 2020.
ISNAD Turhan, Sercan et al. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics 8/1 (April2020), 165-174.
JAMA Turhan S, Kunt M, İşcan İ. On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity. Konuralp J. Math. 2020;8:165–174.
MLA Turhan, Sercan et al. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics, vol. 8, no. 1, 2020, pp. 165-74.
Vancouver Turhan S, Kunt M, İşcan İ. On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity. Konuralp J. Math. 2020;8(1):165-74.
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