Monotonicity of sequences involving convex function and sequence

Let f be an increasing convex (concave, respectively) function defined on [0, 1] and {ai}i∈N be an increasing positive sequence such that { i ( ai ai+1 − 1 )} i∈N decreases ({ i (ai+1 ai − 1 )} i∈N increases, respectively ) , then the sequence { 1 n ∑n i=1 f ( ai an )} n∈N is decreasing. Let f be an increasing convex (concave, respectively) positive function defined on [0, 1] and φ be an increasing convex positive function defined on [0,∞) such that φ(0) = 0 and the sequence { φ(i) [ φ(i) φ(i+1) − 1 ]} i∈N decreases, then the sequence { 1 φ(n) ∑n i=1 f ( φ(i) φ(n) )} n∈N is decreasing. As applications, taking special sequence {ai}i∈N and special functions f and φ, many new inequalities between ratios of means are obtained, and the Alzer’s inequality, the Minc-Sathre’s inequality, and the like, are recovered.

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