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

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 Heikkala et al. (2008) [5]. As an application of this result, we obtain a functional inequality of the generalized modulus.