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.
[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 GAs-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.
[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 GAs-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.
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. Nisan 2020;8(1):165-174.
Chicago
Turhan, Sercan, Mehmet Kunt, ve İmdat İşcan. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics 8, sy. 1 (Nisan 2020): 165-74.
EndNote
Turhan S, Kunt M, İşcan İ (01 Nisan 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, ve İ. İşcan, “On Hermite-Hadamard Type Inequalities with Respect to the Generalization of Some Types of s-Convexity”, Konuralp J. Math., c. 8, sy. 1, ss. 165–174, 2020.
ISNAD
Turhan, Sercan vd. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics 8/1 (Nisan 2020), 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 vd. “On Hermite-Hadamard Type Inequalities With Respect to the Generalization of Some Types of S-Convexity”. Konuralp Journal of Mathematics, c. 8, sy. 1, 2020, ss. 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.