Solution to a conjecture on the Legendre M-function with an application to the generalized modulus

Abstract

Please cite this article in press as: X. Zhang, Solution to a conjecture on the Legendre M -function with an application to the generalized modulus, J. Math. Anal. Appl. (2015), http://dx.Abstract. In this note we give a solution to a conjecture on the Legendre M -function proposed by Heikkala, Linden, Vamanamurthy and Vuorinen in the paper: Generalized elliptic integrals and the Legendre M -function, J. Math. Anal. Appl. 338 (2008), 223-243. As an application of this result, we obtain a functional inequality of the generalized modulus.

Generalized convexity of the inverse hyperbolic cosine function

“…The study of the convexity/concavity with respect to Hölder means, or simply H p;q -convexity/concavity, of special functions has attracted attentions of many researchers, see [2,6,7,[9][10][11][13][14][15][16][17]. In particular, the convexity/concavity of the inverse hyperbolic tangent and sine functions has been studied in [9] and [11], respectively.…”