Redheffer-type Inequalities Research Articles

Some new inequalities for the generalized Fox-Wright functions

<abstract> In this article, we establish the inequalities of the Redheffer-type involving generalized Fox-Wright function. Furthermore, as a consequence, new Redheffer-type inequalities for generalized hypergeometric functions and the four-parametric generalized Mittag-Leffler functions are also discussed by using the suitable values of exponents in generalized inequalities. </abstract>

Geometric and monotonic properties of Ramanujan type entire functions

In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their Mittag–Leffler expansion in such a way that the resulting functions are analytic in the unit disk of the complex plane. A result of Zhang (Proc Am Math Soc 145:241–250, 2017) on the reality of the zeros of Ramanujan type entire functions play important roles in this paper. Moreover, the interlacing properties of the zeros of Ramanujan type functions and its derivative are also useful in the proof of the main results. In addition, by using the Euler–Rayleigh inequalities, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the Ramanujan type entire functions. Finally, we give monotonicity and Redheffer-type inequalities for this function.

Geometric and monotonic properties of hyper-Bessel functions

Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-P\'{o}lya class of real entire functions, intensively.

Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

Extended a constant part of Redheffer's type inequalities

J-L. Li and Y-L. Li \cite{LL2007} gave the following Redheffer's type inequality; \begin{equation*} \frac{ 1 -\left( \frac{x}{\pi} \right)^2 }{ \sqrt{1 + 3 \left( \frac{x}{\pi} \right)^4}} &gt; \frac{\sin{x}}{x} \end{equation*} holds for $0 &lt; x &lt; \pi$, where the constant $3$ is the best possible. In this paper, we establish two inequalities extended the constant part of the above inequality.

Functional inequalities for the Wright functions

ABSTRACTIn this paper, our aim is to show some mean value inequalities for the Wright function, such as Turán-type inequalities, Lazarević-type inequalities, Wilker-type inequalities and Redheffer-type inequalities. Moreover, we prove monotonicity of ratios for sections of series of Wright functions, the result is also closely connected with Turán-type inequalities. In the end of the paper, we present some other inequalities for the Wright function.

Modified Dini functions: monotonicity patterns and functional inequalities

We deduce some new functional inequalities, like Tur\'an type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.

Sharpening Redheffer-type inequalities for circular functions

In this note, some new sharpened Redheffer-type inequalities involving circular functions are established.

Six new Redheffer-type inequalities for circular and hyperbolic functions

In this paper, six new Redheffer-type inequalities involving circular functions and hyperbolic functions are established.