Turán type inequalities for generalized complete elliptic integrals

In this paper our aim is to establish some Turán type inequalities for Gaussian hypergeometric functions and for generalized complete elliptic integrals. These results complete the earlier result of P. Turán proved for Legendre polynomials. Moreover we show that there is a close connection between a Turán type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind. At the end of this paper we prove a recent conjecture of T. Sugawa and M. Vuorinen related to estimates of the hyperbolic distance of the twice punctured plane. Dedicated to my son Koppány.     

Integrals involving complete elliptic integrals

We give a closed-form evaluation of a number of Erdélyi-Kober fractional integrals involving elliptic integrals of the first and second kind, in terms of the ₃F₂ generalized hypergeometric function. Reduction formulae for ₃F₂ enable us to simplify the solutions for a number of particular cases.     

Inequalities and bounds for generalized complete elliptic integrals

Computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained. Also, bounds for some combinations and products for integrals under discussion are established. It has been proven that both families of integrals are logarithmically convex as functions of the first parameter. This property has been employed to obtain several inequalities involving integrals in question.     

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

Bounds for complete elliptic integrals of the first kind

In this note by using some elementary computations we present some new sharp lower and upper bounds for the complete elliptic integrals of the first kind. These results improve some known bounds in the literature and are deduced from the well-known Wallis inequality, which has been studied extensively in the last 10 years.     

Some inequalities of Qi type for double integrals

In the paper, the authors establish some new inequalities of Qi type for double integrals on a rectangle, from which some known integral inequalities of Qi type may be derived.     

A functional identity involving elliptic integrals

We show that the following double integral

$$\begin{aligned} \int _{0}^\pi \mathrm {d}\, x\int _0^x\mathrm {d}\, y\frac{1}{\sqrt{1-\smash [b]{p}\cos x}\sqrt{1+\smash [b]{q\cos y}}} \end{aligned}$$

remains invariant as one trades the parameters p and q for \(p'=\sqrt{1-p^2}\) and \(q'=\sqrt{1-q^2}\), respectively. This invariance property is suggested from symmetry considerations in the operating characteristics of a semiconductor Hall effect device.

     

Sharp inequalities for the generalized elliptic integrals of the first kind

The Ramanujan Journal - Elliptic integrals are of cardinal importance in mathematical analysis and in the field of applied mathematics. Since they cannot be represented by the elementary...     

Quasilinear elliptic inequalities on complete Riemannian manifolds

We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented.     

Algorithms for computing complete elliptic integrals and some related functions

We propose some new algorithms for computing the complete elliptic integrals of the first and second kinds and some related functions. The algorithms are constructed from rapidly converging power series; the sign-definiteness of the terms of the series guarantees their good conditionality (stability with respect to rounding errors). The algorithms turned out flexible and easily adjustable to every specific demand of computational mathematics.     

Some functional inequalities

LetM, N, O be open subsets of ? ⁿ and letF:M×N→O,f:O→?,g: M→?,h: N→? be functions, satisfying the functional inequality $$\forall (x,y) \in M \times N:f[F(x,y)] \leqslant g(x) + h(y).$$ IfF belongs to a certain extensive class of functions, we prove in this note, thatf is bounded above on every compact subset of ? ⁿ , wheneverh is bounded above on a Lebesgue-measurable set of positive Lebesgue-measure, contained inN (no assumptions aboutg are needed). Moreover we give applications of this theorem to generalized convex and subadditive functions.     

Some aspects of elliptic variational inequalities

In this paper we study an existence and the approximation of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We discuss convergence results and give an error estimate for the difference of the two solutions in an appropriate norm. Also, we present some computational results by using fixed point method.     

Some inequalities on general Lp-mixed brightness integrals

Analysis Mathematica - In this paper, we investigate general Lp-mixed brightness integrals which were defined by Yan and Wang. We give their extremal values and establish Brunn-Minkowski type...     

Calculation of the complete elliptic integrals with complex modulus

Summary This is an Addendum to a preceding paper of Morita and Horiguchi [Numer. Math.20, 425–430 (1973)]. Attention is called to an error in the algol procedure given in that paper. A corrected procedure of calculating the complete elliptic integrals of the first and the second kind with complex modulusk is presented, in the form that is itself useful in the calculation of their analytic continuations over the branch cuts.