. If we put m = s = 1 and (, ) = – ity (4.two) in [27]. in Theorem
. If we put m = s = 1 and (, ) = – ity (4.two) in [27]. in Theorem 12, then we receive inequal-GLPG-3221 CFTR Corollary 7. If we place m = n = s = 1 and (, ) = – Corollary 2 in [27].in Theorem 12, then we obtainTheorem 13. Suppose A R is GNE-371 Cancer definitely an open invex subset with respect to : A A R and , A with + (, ) , q 1. Suppose that : A R can be a differentiable function such that L[ + (, ), ]. If | | is often a generalized s-type m reinvex function on [ + (, ), ], then for [0, 1] and s [0, 1], the following inequality + ( + (, )) 1 – 2 (, )+ (, )( x )dx1 q| (, )| 1 two 2qn1- two qi =n(i2 + 3i + two)2i – (1 + 2i i )2si 2i+1 (i + 1)(i + two)A q mi | (q )| , | |q , miholds, where A(.,.) is definitely the arithmetic mean.Axioms 2021, ten,15 ofProof. Let , A . Given that A is an invex set with respect to , for any [0, 1], we have + (, ) A . Suppose that q 1. Applying Lemma 3, the power imply inequality, the generalized s-type m reinvexity of | |q , and the properties of modulus, we’ve + ( + (, )) 1 – 2 (, )+ (, )( x )dx(, )1(1 – 2 ) ( + (, ))d1 0 1- 1 q| (, )| two |1 – two |d1- 1 q1|1 – 2 || ( + (, ))| dq1 q| (, )| 1 2 2qn|1 – two |i =1 1- 1 q[1 – (s )) ]| (in)| + [1 – (s(1 -)) ]| ()| dq i q i =n1 q| (, )| 1 two 2qn1(| |q =|1 – two | [1 – (s ))i ]d +i =1 n 1- two qn1|1 – two | mi | (i =1 qnq )| [1 – (s(1 -))i ]d mi q )| , | |q . mi1 q| (, )| 1 two 2qni =(i2 + 3i + 2)2i – (1 + 2i i )2si 2i+1 (i + 1)(i + 2)A q mi | (Additionally, for q = 1, Applying exactly the same procedure step by step as in Theorem 11, we’re led to the necessary outcome. Corollary eight. If we place n = m = 1 and s = 1 in Theorem 13, then + ( + (, )) 1 – two (, )+ (, )( x )dx(, ) 1 A q | ()|q , | |q .Corollary 9. If we put s = 1 and (, ) = – in [27].in Theorem 13, we get inequality (four.three)Corollary ten. If we place n = 1, s = 1, and (, ) = – Corollary 4 in [27].in Theorem 13, then we obtainTheorem 14. Let A R be an open invex subset with respect to : A A R and , A with + (, ) , q 1, 1 + 1 = 1. Suppose that : A R is often a differentiable p q function such that L[ + (, ), ]. If | | is a generalized s-type m reinvex function on [ + (, ), ], then for [0, 1] and s [0, 1], the following inequality + ( + (, )) 1 – 2 (, )+ (, )( x )dx| (, )| 1 q two( p + 1) two n i1 pn n 2 i + 2 – 2si i + 3i + 2 – 2si m | ( i )|q + | |q 2( i + 2) 2(i + 1)(i + 2) m i =1 i =1 i1 q1 qn two n i + 3i + 2 – 2si i + two – 2si + m | ( i )|q + | |q two(i + 1)(i + two) 2( i + 2) m i =1 i =,Axioms 2021, 10,16 ofholds. Proof. Suppose that , A . Since A is definitely an invex set with respect to , for any [0, 1], we’ve got + (, ) A . From Lemma three, H der scan integral inequality, the generalized s-type m reinvexity of | |q , as well as the properties of modulus, we have + ( + (, )) 1 – 2 (, )+ (, )( x )dx| (, )|1|1 – two || ( + (, ))|d| (, )| 2 | (, )| +0 1(1 -)|1 – 2 | d|1 – 2 | d1 pp1 p1(1 -)| ( + (, ))| dq1 qq1 qp1 p1| ( + (, ))| d| (, )| 1 two( p + 1) 2qnq i =1| | +n(1 -)(1 – (s ) )d +i1 pni =1(1 -)m | ( i )|q [1 – (s(1 -))i ]d mi1 q| (, )| 1 2( p + 1) 2qnq i =1| |n(1 – (s ) )d +i1 pni =1m | ( i )|q [1 – (s(1 -))i ]d mi1 q| (, )| 1 1 2( p + 1) q n 2 + mi | (n n two i + 2 – 2si i + 3i + two – 2si m | ( i )|q + | |q two( i + 2) two(i + 1)(i + 2) m i =1 i =1 i1 q1 qn q n i2 + 3i + 2 – 2si i + 2 – 2si )| + | |q i two(i + 1)(i + 2) 2( i + 2) m i =1 i =.This completes the proof on the desired outcome. Corollary 11. If we put n = m = 1 and s = 1 in Theorem 14, then + ( + (, )) 1 – two (, )+ (, )( x )dx1 q| (, )|1 p+1 p| ()|q two| |q + 3+2| ()|q | |q + 31 q.Corollary 1.
