关于(h,m)-凸函数乘积的Hadamard-型不等式及应用(英文)

作者:孙文兵; 刊名:中国科学院大学学报 上传者:陈志雄

【摘要】建立一些关于(h,m)-凸函数乘积的新Hadamard-型不等式,得到的结果是对通常凸性、第2种意义下的s-凸性、m-凸性、h-凸性意义下的Hadamard-型不等式的推广.

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In recent years,the concept of convex functionDefinition 1.1 Let h:J瓗→瓗be a non-has been extended by some scholars.For example,negative function.We say that f:[0,b]→瓗is anBreckner[1]introduced the concept of s-convexity,(h,m)-convex function with m∈[0,1],if f isand Varo2anec[2]defined h-convex functions.Somenon-negative and for all x,y∈[0,b]andα∈(0,results for Hadamard-type inequalities related to the1),we haveextended convex functions have been obtained[3-7].f(αx+m(1-α)y)≤h(α)f(x)+1 Background knowledgemh(1-α)f(y).If the above inequality is reversed,f is said toIn 2011,zdemir et al.[8]presented the(h,be(h,m)-concave function on[0,b].m)-convex function as follows. Remark 1.1Theorem 1.3 Let f:[0,∞)→瓗be an m-1)if we choose m=1,we have h-convexconvex function with m∈(0,1].If 0≤a<b<∞functions;and f∈L1[a,b],one has the inequality2)if we choose m=1 and h(α)=α,we obtain1b≤b-a∫f(x)dtnon-negative ordinary convex functions;a3)if we choose m=1 and h(α)=αs,we have{af(a)+mf(b)f(b)+mf()minm,m22}.(4)s-convex functions in the second sense;4)if we choose h(α)=α,we have m-convexSome inequalities of Hadamard-type related tofunctions.this new class of(h,m)-convex functions areOne of important applications of the concept ofgiven[8].convex function is the Hadamard’s inequality asTheorem 1.4 Let f:[0,∞)→瓗be an(h,follows.m)-convex function with m∈(0,1]and t∈[0,1].Let f:I瓗→瓗be a convex function and a,bIf 0≤a<b<∞and f∈L1[a,b],the inequality∈I with a<b,then the inequality1bbx)dt≤b-a∫f(f(a+b)1≤≤f(x)

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