Related papers: Bernoulli inequality and hypergeometric functions
In 2007, \'A. Baricz put forward a conjecture concerning Tur\'an-type inequalities for Gaussian hypergeometric functions (see Conjecture \ref{ConjA} in Section \ref{Sec1}). In this paper, the authors disprove this conjecture with several…
The aim of this paper is to present some new Fejer-type results for convex functions. Improvements of Young's inequality (the arithmetic-geometric mean inequality) and other applications to special means are pointed as well.
Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that…
If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…
A version of the recent functional inequality between the Hessians of the square root and the logarithm of positive functions is proven in spaces with non-zero curvature.
We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are…
In this paper we study properties of hyperholomorphic functions on commutative finite algebras. It is investigated the Cauchy-Riemann type conditions for hyperholomorphic functions. We prove that a hyperholomorphic function on a commutative…
The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or…
In this paper, new sharpened Huygens type inequalities involving Bessel and modified Bessel functions of the first kinds are established
In this paper, our aim is to show some mean value inequalities for the Fox-Wright functions, such as Tur\'an--type inequalities, Lazarevi\'c and Wilker--type inequalities. As applications we derive some new type inequalities for…
In this present paper, we establish the log-convexity and Tur\'an type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Tur\'an type inequalities for extended $(p,q)$-confluent…
The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and…
In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the…
We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms.…
In this papaer, we put forward some new definitions and integral inequalities by using fairly elementary analysis.
Some integral properties of Jack polynomials, hypergeometric functions and invariant polynomials are studied for real normed division algebras.
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
We prove an asymptotic formula for $F\left(1/4-it+ir,1/4-it-ir,1/2;x \right)$ as $r, t\to\infty$ and $\alpha=r/t\to0.$ This special case of the Gauss hypergeometric function appears in the explicit formula for the first moment of Maass form…
In this article we derive some polynomial inequalities for Mertens functions.
In this paper, some new inequalities of Ostrowski type established for the class of m- and (alpha,m)-geometrically convex functions which are generalizations of geometric convex functions.