Abstract
We present the best possible lower and upper bounds for the Neuman‐Sándor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means.
Highlights
B > 0 with a / b, the Neuman-Sandor mean M a, b 1 is defined byM a, b a−b 2 sinh−1 a − b / a b1.1 where sinh−1 x√ log x 1 x2 is the inverse hyperbolic sine function.Recently, the theory of bivariate means have been the subject of intensive research 2–17
We present the best possible lower and upper bounds for the Neuman-Sandor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means
The theory of bivariate means have been the subject of intensive research 2
Summary
B > 0 with a / b, the Neuman-Sandor mean M a, b 1 is defined byM a, b a−b 2 sinh−1 a − b / a b1.1 where sinh−1 x√ log x 1 x2 is the inverse hyperbolic sine function.Recently, the theory of bivariate means have been the subject of intensive research 2–17. Many remarkable inequalities for the Neuman-Sandor mean M a, b can be found in the literature 1, 18–20 . In 1, 18 , Neuman and Sandor proved that the double inequalities Neuman 20 proved that the double inequalities αQ a, b 1 − α A a, b < M a, b < βQ a, b 1 − β A a, b , 1.5
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