Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function

Theorem 1.8 (See [75]) Let μ,α,j,γ ,c ∈C, R(μ),R(α),R(j) > 0, R(c) > R(γ ) > 0, p ≥0, δ > 0, 0 < κ ≤δ +R(μ), f ∈L1[a,b] and x ∈[a,b]. Then the extended generalized fractional integral operators ϵγ ,δ,κ,c μ,α,j,w,a+f and ϵγ ,δ,κ,c μ,α,j,w,b–f can be defined by ϵγ ,δ,κ,c μ,α,j,w,a+f (x;p) =  x a (x – t)α–1Eγ ,δ,κ,c μ,α,j  w(x – t)μ;p

Lemma 2.1 Let 0 ≤a < mb and f : [a,mb] →R be a differentiable function such that (ef )′ ∈L1[a,mb]. Then one has

Corollary 2.2 Let 0 ≤a < b and f : [a,b] →R be a differentiable exponential function such that (ef )′ ∈L1[a,b]. Then the identity for the extended generalized fractional integral operators

Theorem 2.3 Let 0 ≤a < mb and f : [a,mb] →R be a differentiable function such that (ef )′ ∈L1[a,mb]. Then the inequality

Corollary 2.4 Let 0 ≤a < b and f : [a,b] →R be a differentiable function such that (ef )′ ∈ L1[a,b]. Then the inequality

Corollary 2.5 If p = 0 and all the assumptions of Theorem 2.3 are satisfied, then one has

Corollary 2.6 Let j = p = 0, m = 1 and all the assumptions of Theorem 2.3 are satisfied, then we get

Corollary 2.7 If we choose j = p = 0, m = 1, α = μ/k and w(s) = 1, then we have the new result

Corollary 2.8 If j = p = 0, m = 1, α = μ k , w(s) = 1 and α = μ, then the inequality

Theorem 2.9 Let 0 ≤a < mb, q,r > 1 such that 1/q + 1/r = 1, and f : [a,mb] →R be a differentiable function such that (ef )′ ∈L1[a,mb]. Then the inequality

Corollary 2.10 Let 0 ≤a < b, p,q > 1 such that 1/p + 1/q = 1, and f : [a,b] →R be a differentiable exponentially convex function such that f ′ ∈L1[a,b]. Then

Corollary 2.11 If we set p = 0, then we get the inequality

Corollary 2.12 If we set j = p = 0 and m = 1, then one has

Corollary 2.13 If we set j = p = 0, m = 1 and α = μ k , then we have

Corollary 2.14 If we set j = p = 0, m = 1 and w(s) = 1, then

Corollary 2.15 If we set j = p = 0, m = 1, w(s) = 1 and α = 1, then

Theorem 2.16 Let 0 ≤a < mb, f : [a,mb] −→R be an exponentially m-convex such that f ∈L1[a,mb]. Then the inequalities for extended generalized fractional integral operators 2ef ( a+mb 2 )ζα,( a+mb 2 )+(mb;p) ≤  εγ ,δ,κ,c μ,α,j,w′,( a+mb 2 )+ef (mb;p) + mα+1 εγ ,δ,κ,c

Corollary 2.17 Let 0 ≤a < b and f : [a,b] →R be an exponentially convex function such that f ∈L1[a,b]. Then the inequalities for extended generalized fractional integral opera- tors 2ef ( a+b 2 )ζα,( a+b 2 )+(b;p) ≤  εγ ,δ,κ,c μ,α,j,w′,( a+b 2 )+ef (b;p) +  εγ ,δ,κ,c μ,α,j,w′,( a+b

Corollary 2.18 If we set p = 0, then we have the following inequalities: 2ef ( a+mb 2 ) εγ ,δ,κ μ,α,j,w′,( a+mb 2 )+1