Optimal bounds for the sine and hyperbolic tangent means. IV
The authors provide optimal bounds for the sine and hyperbolic tangent means in terms of various weighted means of the arithmetic and centroidal means. The sine and hyperbolic tangent means are defined, respectively, as follows: \[ M_{\sin} (x,y)= \left\{ \begin{array}{lcl} \frac{x-y}{2\sin\left(\frac{x-y}{x+y}\right)} & \mbox{if} & x\neq y \\ x & \mbox{if} & x=y \end{array}\right. \] and \[ M_{\tanh} (x,y)= \left\{ \begin{array}{lcl} \frac{x-y}{2\tanh\left(\frac{x-y}{x+y}\right)} & \mbox{if} & x\neq y \\ x & \mbox{if} & x=y \end{array}\right.. \] For Part I and II see [the authors, ``Optimal bounds for the sine and hyperbolic tangent means'', J. Math., Punjab Univ. 52, No. 5, 77--88 (2020); J. Appl. Anal. 27, No. 1, 65--72 (2021; Zbl 1473.26042)].
scientific article; zbMATH DE number 7390106
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