Refinements and Generalizations of Some Fractional Integral Inequalities via Strongly Convex Functions

Abstract

In this article, we have established the Hadamard inequalities for strongly convex functions using generalized Riemann–Liouville fractional integrals. The findings of this paper provide refinements of some fractional integral inequalities. Furthermore, the error bounds of these inequalities are given by using two generalized integral identities.

Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially α , h − m -

“…In Section 3, error estimations of fractional Hadamard inequality for strongly exponentially (α, h − m) are obtained with the help of two fractional integral identities. e outcomes of this article are connected with already established results given in [15,16,19,20,[26][27][28][29][30][31][32][33][34][35][36][37].…”

“…(i) If we take λ � 0 in (60), then the inequality stated in eorem 11 in [38] is obtained (ii) If we take η � 0 and k � 1 in (60), then the inequality stated in eorem 8 in [37] is obtained (iii) If we take η � 0 and α � 1 in (60), then the inequality stated in Corollary 12 in [37] is obtained (iv) If we take h(t) � t, k � 1, η � 0, α � 1, λ � 0, and ψ as the identity function in (60), then the inequality stated in eorem 2.7 in [30] is obtained (v) If we take η � 0, λ � 0, h(t) � t, α � 1, m � 1, and ψ as the identity function in (60), then the inequality stated in eorem 3.2 in [20] is obtained (vi) If we take k � 1, h(t) � t, m � 1, η � 0, ξ � 1, α � 1, λ � 0, and ψ as the identity function in (60), then the inequality stated in eorem 2.4 in [33] is obtained (vii) If we take η � 0, α � 1, and h(t) � t s in (60), then the inequality stated in Corollary 9 in [35] is obtained (viii) If we take α � 1, m � 1, h(t) � t, and η � 0 in (60), then the inequality stated in eorem 14 in [32] is obtained Now, we give inequality (60) for strongly exponentially (h − m)-convex, strongly exponentially (s, m)-convex, strongly exponentially m-convex, and strongly exponentially convex functions. e η a/m 2 ( )+b ( )…”

“…η � 0, λ � 0, and ψ as the identity function in ( 17), then Hadamard inequality is obtained (v) If we take η � 0, m � 1, α � 1, λ � 0, and h(t) � t in ( 17), then the inequality stated in eorem 1 in [26] is obtained (vi) If we take η � 0, m � 1, α � 1, and h(t) � t in ( 17), then the inequality stated in eorem 10 in [32] 17), then the inequality stated in eorem 2.1 in [34] is obtained (viii) If we take α � 1, k � 1, h(t) � t, η � 0, λ � 0, and ψ as the identity function in (17), then the inequality stated in eorem 2.1 in [31] is obtained (ix) If we take α � 1, λ � 0, h(t) � t s , and ψ as the identity function in (17), then the inequality stated in eorem 2 in [36] is obtained (x) If we take α � 1, η � 0, λ � 0, and h(t) � t s in ( 17), then the inequality stated in Corollary 1 in [35] is obtained (xi) If we take η � 0 and k � 1 in ( 17), then the inequality stated in eorem 4 in [37] is obtained (xii) If we take η � 0, k � 1, and α � 1 in ( 17), then the inequality stated in Corollary 1 in [37] is obtained (xiii) If we take λ � 0 in (17), then the inequality stated in eorem 7 in [38] is obtained Now, we give inequality ( 17) for strongly exponentially (h − m)-convex, strongly exponentially (s, m)-convex, strongly exponentially m-convex, and strongly exponentially convex functions.…”

“…and ψ as the identity function in (31), then the Hadamard inequality is obtained (vi) If we take h(t) � t, m � 1, α � 1, and η � 0 in (31), then the inequality stated in eorem 11 in [32] is obtained (vii) If we take α � 1, h(t) � t, k � 1, λ � 0, η � 0, and ψ as the identity function in (31), then the inequality stated in eorem 2.1 in [30] is obtained (viii) If we take λ � 0, α � 1, h(t) � t s , and η � 0 in (31), then the inequality stated in Corollary 3 in [35] is obtained (ix) If we take λ � 0 in (31), then the inequality stated in eorem 8 in [38] is obtained Now, we give inequality (31) for strongly exponentially (h − m)-convex, strongly exponentially (s, m)-convex, strongly exponentially m-convex, and strongly exponentially convex functions.…”

“…(i) If we take η � 0 and k � 1 in (45), then the inequality stated in eorem 6 in [37] is obtained (ii) If we take η � 0 and α � 1 in (45), then the inequality stated in Corollary 7 in [37] is obtained (iii) If we take h(t) � t, m � 1, α � 1, and η � 0 in (45), then the inequality stated in eorem 12 in [32] is obtained (iv) If we take m � 1, α � 1, η � 0, λ � 0, and h(t) � t s in (45), then the inequality stated in eorem 2 in [26] is obtained (v) If we take m � 1, α � 1, λ � 0, h(t) � t, η � 0, and ψ as the identity function in (45), then eorem 6 is obtained…”