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Register a new userApplications of a duality between generalized trigonometric and hyperbolic functions
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It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinovi{c}-Adamovi{c}-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.
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Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.
With respect to generalized trigonometric functions, since the discovery of double-angle formula for a special case by Edmunds, Gurka and Lang in 2012, no double-angle formulas have been found. In this paper, we will establish new double-angle formulas of generalized trigonometric functions in two special cases.
Famous Redheffers inequality is generalized to a class of anti-periodic functions. We apply the novel inequality to the generalized trigonometric functions and establish several Redheffer-type inequalities for these functions.
The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.
Using the Fourier analysis techniques on hyperbolic spaces and Greens function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $frac{n-1}{2}$-th order Hardy-Sobolev-Mazya inequality in the upper half space of dimension $n$ coincides with the best $frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $ngeq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Mazya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Greens functions of the operator $ -Delta_{mathbb{H}}-frac{(n-1)^{2}}{4}$ on the hyperbolic space $mathbb{B}^n$ and operators of the product form are given, where $frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Delta_{mathbb{H}}$ on $mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Greens function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).
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