Functional Inequalities and Monotonicity Results for Modified Lommel Functions of the First Kind

We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Turán type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.

On generalized Bessel–Maitland function

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by $\mathcal{J}_{\nu , \lambda }^{\mu }$ J ν , λ μ , where $\mu >0$ μ > 0 and $\lambda ,\nu \in \mathbb{C\ }$ λ , ν ∈ C get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions $\mathcal{I}_{\nu ,\lambda }^{\mu }$ I ν , λ μ defined by $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$ I ν , λ μ ( z ) = i − 2 λ − ν J ν , λ μ ( i z ) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$ I ν , λ μ ( z ) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

Extended elliptic-type integrals with associated properties and Turán-type inequalities

Our aim is to study and investigate the family of $(p, q)$ ( p , q ) -extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with $(p, q)$ ( p , q ) -extended Gauss’ hypergeometric function and $(p, q)$ ( p , q ) -extended Appell’s double hypergeometric function $F_{1}$ F 1 . Turán-type inequalities including log-convexity properties are proved for these $(p, q)$ ( p , q ) -extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these $(p, q)$ ( p , q ) -extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with $(p, q)$ ( p , q ) -extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce $(p, q)$ ( p , q ) -extension of the Epstein–Hubbell (E-H) elliptic-type integral.

Sampling the Flow of a Bandlimited Function

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states $$f_t=\phi _t*f$$ f t = ϕ t ∗ f resulting from the convolution with a kernel $$\phi _t$$ ϕ t . It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $$\widehat{f}$$ f ^ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

Several Turán-Type Inequalities for the Generalized Mittag-Leffler Function

In this paper, several Turán-type inequalities for the more generalized Mittag-Leffler function are proved. In addition, we also gave affirmative answers to two open problems posed by Mehrez and Sitnik.

On Turan-type inequalities for trigonometric polynomials of half-integer order

Some exact inequalities of the Turan type are obtained in the paper for trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, such that all $2n+1$ their zeros are real and located on a segment $[0;2\pi )$. Namely, the inequality that relates the norms in the space $C$ of the trigonometric polynomials $h(x)$ of half-integer order $n+\frac {1}{2}$, $n=1, 2, ...$, and its second derivative $h''(x)$, $\|h''\|_c\ge \frac {2n+1}{4}\|h\|_c$, that is the inequalities that connect the norms in the space $L_2$ of the trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, and its first derivative $h'(x)$, that is $\|h'\|_{L_2}\ge \sqrt {\frac {2n+1}{8}}\|h\|_{L_2}$. The resulting inequalities cannot be improved. In proving the theorems, we use the method that was developed by V.F. Babenko and S.A. Pichugov for trigonometric polynomials, all of whose roots are real.