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We find the greatest value p and least value q such that the double inequality Lp(a, b) < T(a, b) < Lq(a, b) holds for all a, b > 0 with a = b, and give a new upper bound for the complete elliptic integral of the second kind. Here T (a, b) = 2 π π/2 0 a 2 cos 2 θ + b 2 sin 2 θdθ and Lp(a, b) = (a p+1 + b p+1 )/(a p + b p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively. Mathematics Subject Classification (2010). Primary 26E60.

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In the article, we provide a sufficient condition for value range of the constant c such that the functionis strictly concave on (0,1) for a ∈ (0,1/2] , which generalize a very recently obtained result that the functionis strictly concave on (0,1) if and only if c = e 4/3 . As applications, we present new bounds for, where K a (x) is the generalized elliptic integral of the first kind and K (x) = K 1/2 (x) .

In the article, we establish several new inequalities for the generalized trigonometric and hyperbolic functions with one parameter, generalize the well known Mitrinović-Adamović, Lazarević, Huygens-type, Wilker-type and Cusa-Huygens-type inequalities to the cases of the generalized trigonometric and hyperbolic functions with one parameter.