Related papers: The Inequalities for Some Types of $q$-Integrals
Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in…
Certain new inequalities for the sums of factorials are presented.
In this paper, we prove the finiteness of the number of integer solutions of the decomposable form inequalities. We also study the number of integer solutions of a sequence of decomposable form inequalities.
Some q-analogues of classical integral transforms have recently been investigated by many authors in diverse citations. The q-analogues of the Natural transform are not known nor used. In the present paper, we are concerned with definitions…
In this paper, we establish some new integral inequalities for $(\alpha, m)-$convex functions and quasi-convex functions, respectively. Our results in special cases recapture known results.
For real numbers $p,q>1$ we consider the following family of integrals: \begin{equation*} \int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad \mbox{and}\quad…
Following an idea due to J. Bernoulli, we explore the q-analogue of the sums of powers of consecutive integers.
This paper is inspired by the work of J. S\'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $ \frac{\Gamma_{q}(x+1)}{ \Gamma_{q} \left( x+\frac{1}{2}\right)}$, where $\Gamma_{q}(x)$ is the…
We show some new Wolstenholme type $q$-congruences for some classes of multiple $q$-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are $q$-extensions of the corresponding…
In this paper, authors study the generalized complete $(p,q)$-elliptic integrals of the first and the second kind as an application of generalized trigonometric functions with two parameters, and establish the Tur\'an type inequalities of…
Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using…
Earlier work introduced a method for obtaining indefinite $q$-integrals of $q$-special functions from the second-order linear $q$-difference equations that define them. In this paper, we reformulate the method in terms of $q$-Riccati…
Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q tends to…
Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces.
In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for…
We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial…
We define and study the interpolated finite multiple harmonic $q$-series. A generating function of the sums of the interpolated finite multiple harmonic $q$-series with fixed weight, depth and $i$-height is computed. Some Ohno-Zagier type…
In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{\mu,\nu}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality,…
We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.
In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the…