Bounds of a unified integral operator for (<i>s</i>,<i>m</i>)-convex functions and their consequences

Refinements of Some Integral Inequalities fors,m-Convex Functions

“…Remark 1 (i) If we consider λ � 0 in (27), then eorem 3.1 in [32] can be obtained, and for λ > 0, we get its refinement (ii) If we consider ϕ(t) � t α and g(x) � x in (27), then eorem 1 can be obtained (iii) If we consider s � m � 1 in the result of (ii), then Corollary 1 in [31] can be obtained (iv) If we consider α � β in the result of (ii), then Corollary 3 in [31] can be obtained (v) If we consider f ∈ L ∞ [a, b] in the result of (ii), then Corollary 5 in [31] can be obtained (vi) If we consider α � β in the result of (v), then Corollary 7 in [31] can be obtained (vii) If we consider s � 1 in the result of (ii), then Corollary 5 in [31] can be obtained (viii) If we consider (s, m) � (1, 1) in (27), then eorem 2 in [33] is obtained (ix) If we consider α � β, λ � 0, and (s, m) � (1, 1) in (27), then eorem 8 in [23] is obtained (x) If we consider λ � 0 and p � ω � 0 in (27), then eorem 1 in [34] is obtained (xi) If we consider λ � 0, ϕ(t) � Γ(α)t α , p � ω � 0, and (s, m) � (1, 1) in ( 27), then eorem 1 in [35] is obtained (xii) If we consider α � β in the result of (xi), then Corollary 1 in [35] is obtained (xiii) If we consider λ � 0, ϕ(t) � t α , g(x) � x, and m � 1 in (27), then eorem 2.1 in [36] is obtained (xvi) If we consider α � β in the result of (xiii), then Corollary 2.1 in [36] (27), then eorem 1 in [38] is obtained (xviii) If we consider α � β in the result of (xvii), then Corollary 1 in [38] can be obtained (xviii) If we consider α � β � 1 and x � a or x � b in the result of (xvii), then Corollary 2 in [38] can be obtained (xix) If we consider α � β � 1 and x � ((a + b)/2) in the result of (xvii), then Corollary 3 in [38] can be obtained e following lemma is very helpful in the proof of the upcoming theorem, see [31].…”

“…Remark 3 (i) If we consider λ � 0 in (52), then eorem 3.4 in [32] can be obtained (ii) If we consider ϕ(t) � t α and g(x) � x in (52), then eorem 6 in [31] can be obtained (iii) If we consider s � m � 1 in the result of (ii), then Corollary 13 in [31] can be obtained (iv) If we consider α � β in the result of (ii), then Corollary 11 in [31] can be obtained (v) If we consider (s, m) � (1, 1) in (52), then eorem 3 in [33] is obtained (vi) If we consider λ � 0 and (s, m) � (1, 1) in (52), then eorem 25 in [23] is obtained (vii) If we consider λ � 0 and p � ω � 0 in (52), then eorem 2 in [34] is obtained (viii) If we consider λ � 0, ϕ(t) � Γ(α)t α+1 , p � ω � 0, and (s, m) � (1, 1) in (52), then eorem 2 in [35] is obtained (ix) If we consider α � β in the result of (viii), then Corollary 2 in [35] is obtained (x) If we consider λ � 0, ϕ(t) � t α , g(x) � x, and m � 1 in (52), then eorem 2.3 in [36] is obtained (xi) If we consider α � β in the result of (x), then Corollary 2.5 in [36] is obtained (xii) If we consider λ � 0, ϕ(t) � Γ(α)t α/k+1 , (s, m) � (1, 1), g(x) � x, and p � ω � 0 in (52), then eorem 2 in [37] can be obtained (xiii) If we consider α � β in the result of (xii), then Corollary 4 in [37] can be obtained (xiv) If we consider α � β � k � 1 and x � ((a + b)/2) in the result of (xii), then Corollary 5 in [37] can be obtained (xv) If we consider λ � 0, ϕ(t) � Γ(α)t α+1 , g(x) � x, p � ω � 0, and (s, m) � (1, 1) in ( 52), then eorem 2 in [38] is obtained (xvi) If we consider α � β in the result of (xv), then Corollary 5 in [38] can be obtained…”

“…In [32], we studied the properties of the kernel given in (13). Here, we are interested in the following property.…”

Inequalities for Unified Integral Operators via Stronglyα,h‐m-Convexity

“…If ξ and ϕ/I are of opposite monotonicities, then (18) holds in reverse direction. For further properties, see [37].…”

On Generalized Strongly Convex Functions and Unified Integral Operators

“…e reverse of inequality (10) holds when g and (ϕ/I) are of opposite monotonicity. For further properties, see [27].…”