Abstract
In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function.
Highlights
Let p, q ∈ R and a, b > with a = b
It is well known that Sp,q(a, b) is continuous and symmetric on the domain {(p, q, a, b) : p, q ∈ R, a >, b > } and strictly increasing with respect to its parameters p, q ∈ R for fixed a, b > with a = b
Many bivariate means are particular cases of the Stolarksy mean, and many remarkable inequalities and properties for this mean can be found in the literature
Summary
Let b > a > and t = log b/a ∈ ( , ∞). ). the function t → [log Hp,q(t)]/t is strictly decreasing (increasing) from ( , ∞) onto ( , (p + q)/ ) (((p + q)/ , )) if p + q > (< ). Proof Let g (t) = [log Hp,q(t)]/t and g (t) = t.
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