一个不等式的进一步推广和加强

A Further Generalization and Sharpening of an Inequality

利用幂级数展式和凸函数的性质把关于一个不等式的推广和强化的两个最新结果推广到更加一般的情形p(p -1 ) d ap- 1pn+1-ap- 1pm <∑nk=m1a1pk<p(p -1 ) d ap- 1pn -ap- 1pm- 1和p2 (p -1 ) d ap- 1pn+2 +ap- 1pn -ap- 1pm+1-ap- 1pm- 1<∑nk=m1a1pk<p2 (p -1 ) d ap- 1pn+1+ap- 1pn -ap- 1pm -ap- 1pm- 1.

Two most recent results on generalization and sharpening of an inequality have been further generalized and sharpened to more general versions by using the expansion of power series and the properties of convex and concave functionsp(p-1)da p-1p n+1 -a p-1p m<∑nk=m1a 1p k<p(p-1)da p-1p n-a p-1p m-1 andp2(p-1)da p-1p n+2 +a p-1p n-a p-1p m+1 -a p-1p m-1 <∑nk=m1a 1p k<p2(p-1)da p-1p n+1 +a p-1p n-a p-1p m-a p-1p m-1 .