scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

Completely monotonic functions related to the q-gamma and the q-trigamma functions

2015 ◽  
Vol 13 (02) ◽  
pp. 125-134 ◽  
Author(s):  
Ahmed Salem
Keyword(s):  
Gamma Function ◽  
Infinite Class ◽  
Positive Real ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Monotonic Functions ◽  
Polygamma Function ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, two completely monotonic functions involving the q-gamma and the q-trigamma functions where q is a positive real, are introduced and exploited to derive sharp bounds for the q-gamma function in terms of the q-trigamma function. These results, when letting q → 1, are shown to be new. Also, sharp bounds for the q-digamma function in terms of the q-tetragamma function are derived. Furthermore, an infinite class of inequalities for the q-polygamma function is established.

scholarly journals Wiener-type invariants of some graph operations

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 103-113 ◽  
Author(s):  
S. Hossein-Zadeh ◽  
A. Hamzeh ◽  
A.R. Ashrafi
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Harary Index ◽  
Graph Operations ◽  
Type Invariant ◽  
Wiener Type

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, ? a real number, and W?(G) =?k?1 d(G, k)k?. W?(G) is called the Wiener-type invariant of G associated to real number ?. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyper- Wiener index and Tratch-Stankevich-Zefirov index are calculated. Some upper and lower bounds are also presented.

scholarly journals Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Mohammad Reza Farahani ◽  
Shaohui Wang ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Mathematical Properties ◽  
Bicyclic Graphs ◽  
Important Branch

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as∑uv∈E(G)‍(d(u)+d(v))2, whered(v)is the degree of the vertexvin a graphG=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs amongn-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.

scholarly journals Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

2021 ◽  
Author(s):  
Adam Lecko ◽  
Barbara Śmiarowska
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Upper And Lower Bounds ◽  
Toeplitz Determinants ◽  
Sharp Bounds

AbstractSharp upper and lower bounds of the Hermitian Toeplitz determinants of the second and third orders are found for various subclasses of close-to-convex functions.

A NOTE ON SPIRALLIKE FUNCTIONS

2021 ◽  
pp. 1-7
Author(s):  
Y. J. SIM ◽  
D. K. THOMAS
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Open Problem ◽  
Univalent Functions ◽  
Inverse Function ◽  
Upper And Lower Bounds ◽  
Spirallike Functions ◽  
New Inequalities

Abstract Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$ , that is, functions satisfying $$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$ for $z\in \mathbb {D}$ , $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$ . We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$ , thus solving an open problem and presenting some new inequalities for coefficient differences.

scholarly journals On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Hesham Moustafa ◽  
Hanan Almuashi ◽  
Mansour Mahmoud
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Upper And Lower Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

scholarly journals Bounds on integrals with respect to multivariate copulas

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Michael Preischl
Keyword(s):  
Lower Bounds ◽  
Open Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Multidimensional Case ◽  
Assignment Problems ◽  
Dimensional Dependence ◽  
Dependence Measures

AbstractIn this paper, we present a method to obtain upper and lower bounds on integrals with respect to copulas by solving the corresponding assignment problems (AP’s). In their 2014 paper, Hofer and Iacó proposed this approach for two dimensions and stated the generalization to arbitrary dimensons as an open problem. We will clarify the connection between copulas and AP’s and thus find an extension to the multidimensional case. Furthermore, we provide convergence statements and, as applications, we consider three dimensional dependence measures as well as an example from finance.

Sharp upper and lower bounds for the gamma function

2009 ◽  
Vol 139 (4) ◽  
pp. 709-718 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Upper And Lower Bounds ◽  
Image Position

We prove that for all x > 0, we havewith the best possible constants α = 0 and $\beta=\tfrac{1}{1620}$.

scholarly journals Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

2021 ◽  
Vol 7 (2) ◽  
pp. 2529-2542
Author(s):  
Chang Liu ◽  
◽  
Jianping Li
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Randić Index ◽  
Extremal Graphs ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.</p></abstract>

scholarly journals (β ,α)−Connectivity Index of Graphs

2017 ◽  
Vol 2 (1) ◽  
pp. 21-30 ◽  
Author(s):  
B. Basavanagoud ◽  
Veena R. Desai ◽  
Shreekant Patil
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Upper And Lower Bounds ◽  
Line Graphs ◽  
Extremal Graphs ◽  
Acentric Factor ◽  
Wheel Graphs ◽  
Octane Isomers ◽  
Ladder Graphs

AbstractLet Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as$$\begin{array}{} \displaystyle ^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}. \end{array}$$The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

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