We determine the best positive constants p and q such that (sinh x/x) p < x/ sin x < (sinh x/x) q. Some applications for Wilker's type inequalities are given. Mathematics subject classification (2010): 26D05, 26D07, 26D99. Keywords and phrases: inequalities; trigonometric functions; hyperbolic functions; means and their inequalities. R E F E R E N C E S [1] R. KLÉNKL´KLÉN, M. VISURI AND M. VUORINEN, On Jordan type inequalities for hyperbolic functions, J.
This paper deals with thep-version of the Schwab-Borchardt mean. Lower and upper bounds for this mean, expressed in terms of the weighted geometric and arithmetic means of its variables, are obtained. Applications to four bivariate means, introduced earlier by the author of this paper, are included.
Abstract. Inequalities for the Schwab-Borchardt mean are obtained. They contain known results for the trigonometric and hyperbolic functions including those obtained by J. Wilker [15] and C. Huygens [5]. The main results of this paper can also be utilized to obtain new inequalities for some bivariate means including the logarithmic mean and two means introduced by Seiffert.Mathematics subject classification (2010): Primary: 26D05, 26D07, 33B10.
This paper deals with a one-parameter generalization of the Schwab-Borchardt mean. The new mean is defined in terms of the inverse functions of the generalized trigonometric and generalized hyperbolic functions. The four new bivariate means are introduced as particular cases of thep-version of the Schwab-Borchardt mean. For the particular value of the parameterp, these means become either the classical logarithmic mean or the Seiffert means or the Neuman-Sándor mean. Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are also established.
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