Abstract

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by , ; , and , ; , , respectively. In this paper, we find the greatest value and the least value such that the double inequality holds for all with .

Highlights

  • For k ∈ 0, ∞ , the power-type Heron mean Hk real numbers a and b are defined by Hk a, b a, b ak and the Seiffert mean ab k/2 bk /3 1/k, T k a, b / 0; of Hk two a, b pos√itive ab, k 0 and T a, b a − b /2 arctan a − b / a b , a / b; T a, b a, a b, respectively

  • For k ∈ 0, ∞, the power-type Heron mean Hk a, b and the Seiffert mean T a, b of two positive real numbers a and b are defined by Hk a, b⎧ ⎪⎪⎨ ak ⎪⎪⎩√ab, ab k/2 3 bk 1/k, k / 0, k 0, ⎧ T a, b⎪⎨ ⎪⎩a2,arctan a−b a−b / a b, a / b, a b, 1.2 respectively.Journal of Inequalities and ApplicationsRecently, the means of two variables have been the subject of intensive research 1– 15

  • For k ∈ 0, ∞, the power-type Heron mean Hk real numbers a and b are defined by Hk a, b a, b ak and the Seiffert mean ab k/2 bk /3 1/k, T k a, b / 0; of Hk two a, b pos√itive ab, k 0 and T a, b a − b /2 arctan a − b / a b, a / b; T a, b a, a b, respectively

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Summary

Introduction

For k ∈ 0, ∞ , the power-type Heron mean Hk real numbers a and b are defined by Hk a, b a, b ak and the Seiffert mean ab k/2 bk /3 1/k, T k a, b / 0; of Hk two a, b pos√itive ab, k 0 and T a, b a − b /2 arctan a − b / a b , a / b; T a, b a, a b, respectively. Hlog 3/ log π/2 a, b < T a, b < H5/2 a, b , 1.8 and Hlog 3/ log π/2 a, b and H5/2 a, b are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean T a, b , respectively.

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