# Volume 11 (2019), Number 1-2

## Certain Subclass of Meromorphic Functions Involving $q$-Ruscheweyh Differential Operator

### Author(s): ABDULLAH ALSOBOH and MASLINA DARUS

**Abstract:** In this paper, we introduce a new q-analogue of differential operator involving
$q$-Ruscheweyh
operator with a new subclass of meromorphic functions. We obtain coeﬃcient conditions and show some
properties for function $f$ belonging to this subclass such as convolution conditions, closure and convex combinations.
Finally, the neighbourhoods problem is solved.

## A Note on the Bessel Form of Parameter $3\u22152$

### Author(s): BAGHDADI ALOUI and LOTFI KHÉRIJI

**Abstract:** In this manuscript, we consider a certain raising operator (with a nonzero
free parameter) and we prove the following statement: up to normalization, the only orthogonal
sequence that remain orthogonal after application of this raising operator is the one obtained by
dilating the Bessel polynomials of parameter $3\u22152$.

## Existence and Uniqueness Theorems for Fourth-Order Equations With Boundary Conditions

### Author(s): ROWAIDA ALRAJHI and LAZHAR BOUGOFFA

**Abstract:** The purpose of this paper is to prove the existence and uniqueness theorem of the
boundary value problem for fourth-order differential equations
$${u\left(4\right)}^{}+q\left(x\right)u\left(x\right)=g\left(x\right),\phantom{\rule{2.77695pt}{0ex}}0<x<1,$$
subject to the BC: $$u\left(0\right)={u}^{\prime}\left(0\right)={u}^{\u2033}\left(1\right)={u}^{\u2034}\left(1\right)=0$$ in the Sobolev space
${\mathbb{H}}^{4}\left(\left[0,1\right]\right)$ by using an a priori estimate. We also investigate the Schauder’s ﬁxed point theorem for proving the existence theorem of the boundary value problem for
fourth-order nonlinear differential equations $${u}^{4}-f\left(x,{u}^{\prime},{u}^{\prime},{u}^{\u2033},{u}^{\u2034}\right),\phantom{\rule{2.77695pt}{0ex}}0<x<1,$$ under the above boundary conditions (BC),
where $f:\left[0,1\right]\times {\mathbb{R}}^{4}\to \mathbb{R}$ is a continuous function and satisfies
$$\left|f\left(x,u,v,w,z\right)\right|\le a+{a}_{0}\left|u\right|+{a}_{1}\left|v\right|+{a}_{2}\left|w\right|+{a}_{3}\left|z\right|,$$ where
$a,{a}_{i}>0,\phantom{\rule{2.77695pt}{0ex}}i=0,\dots ,3$.

## Hermite-Hadamard-Fejer Type Inequalities for $s-p-$Convex Functions of Several Kinds

### Author(s): AMBREEN ARSHAD and ASIF R. KHAN

**Abstract:** Earlier, Mehmet has done work on Hermite-Hadamard-Fejer type inequality for
$p-$convex functions in [6]. Now we generalized the work of Mehmet for $p-$convexity of Hermite-Hadamard-Fejer type inequalities for
$s-p-$convex
functions of 1st kind and 2nd kind.

## Positive Periodic Solutions for Second-Order Neutral Diﬀerence Equations With Variable Coefficients

### Author(s): ABDELOUAHEB ARDJOUNI and AHCENE DJOUDI

**Abstract:** In this article, we obtain sufficient conditions for the existence of positive
periodic solutions for a second-order neutral difference equation with variable coefficients.
The main tool employed here is the Krasnoselskii fixed point theorem dealing with a sum of two mappings, one is a contraction and the other is a completely continuous.

## Estimation of Upper Bounds Involving Katugampola Fractional Integrals

### Author(s): MUHAMMAD UZAIR AWAN, NOUSHEEN AKHTAR, MUHAMMAD ASLAM NOOR, MARCELA V. MIHAI and KHALIDA INAYAT NOOR

**Abstract:** The main objective of this article is to give some further generalizations of
Hermite-Hadamard type inequalities via harmonic convex functions. The inequalities involve Katugampola’s
fractional integral. We also give new estimation to upper bounds essentially using Katugampola’s
fractional integrals. Special cases are also discussed.

## Generalizations of Opial Type Inequalities in Two Variables Using $p$-Norms

### Author(s): HÜSEYIN BUDAK

**Abstract:** In this paper, using the $p$-norms, we obtain some generalized Opial type inequalities in two
variables for two functions. The results in this paper generalize several inequalities obtained in
earlier works.

## Cubic Spline Approximation of the Reliability Polynomials of Two Dual Hammock Networks

### Author(s): GABRIELA CRISTESCU and VLAD-FLORIN DRĂGOI

**Abstract:** The property of preserving the convexity and concavity of the Bernstein polynomial
and of the Bézier curves is used to generate a method of approximating the reliability polynomial of a
hammock network. The mutual behaviour of the reliability polynomials of two dual hammock networks is
used to generate a system of constraints since the initial information is not enough for using a
classical approximation scheme. A cubic spline function is constructed to generate approximations of
the coefficients of the two reliability polynomials. As consequence, an approximation algorithm is
described and tested through simulations on hammocks with known reliability, comparing the results
with the results of approximations attempts from literature.

## A Note on Numerical Comparison of Some Multiplicative Bounds Related to Weighted Arithmetic and Geometric Means

### Author(s): SILVESTRU SEVER DRAGOMIR and ALASDAIR MCANDREW

**Abstract:** In this note we provided some numerical comparison for the upper and lower bounds
in some recent inequalities related to the famous Young’s inequality for two positive numbers. We drew
the conclusion that neither of the inequalities below is always best.

## Some Discrete Inequalities for Convex Functions Deﬁned on Linear Spaces

### Author(s): SILVESTRU SEVER DRAGOMIR

**Abstract:** In this paper we provide some discrete inequalities related to the Hermite-Hadamard
result for convex functions deﬁned on convex subsets in a linear space. Applications for norms and
univariate real functions with an example for the logarithm, are also given.

## Extension of Results Analogous to Beta Functions by the Way of $\left(\alpha ,s\right)$ Pre Invexity

### Author(s): ZAMIR HUSSAIN and MUHAMMAD MUDDASSAR

**Abstract:** In this note, we establish an extension of existing results analogous to Euler
Beta function link with the integral inequalities of the type of Hermite-Hadamard’s by weaken the
condition of convexity, when the power of the absolute value is
$\left(\alpha ,s\right)$ preinvex mappings.

## On Some Connected Groups of Automorphisms of Weil Algebras

### Author(s): MIROSLAV KUREŠ and JAN ŠÚTORA

**Abstract:** The method of direct calculation of the group of
$\mathbb{R}$-algebra
automorphisms of a Weil algebra is presented in detail. The paper is focused on the case of a
one-componental group and presents two cases of values of the determinant of its linear part.

## Differential Subordinations Defined by Using Sălăgean Integral Operator at the Class of Meromorphic Functions

### Author(s): CAMELIA MĂDĂLINA MARTINESCU and SORIN MIREL STOIAN

**Abstract:** By using the Sălăgean integral operator
${I}^{n}f\left(z\right),\phantom{\rule{2.77695pt}{0ex}}z\in U$, we introduce a general class of holomorphic functions denoted by
${\sum}_{k,m}\left(\alpha ,n\right)$
and we obtain an inclusion relation related to this class and some differential subordinations.

## On Certain Family of Multivalent Harmonic Functions Associated With Jung–Kim–Srivastava Operator

### Author(s): SHAHRAM NAJAFZADEH

**Abstract:** New families of harmonic multivalent functions in terms Jung–Kim–Srivastava
integral operator are introduced. Sufficient and necessary conditions for coefficients are obtained.
Also extreme points, distortion bounds and convex combinations are investigated.

## Exponentially General Convex Functions

### Author(s): MUHAMMAD ASLAM NOOR and KHALIDA INAYAT NOOR

**Abstract:** In this paper, we define and introduce some new classes of the exponentially convex
functions involving an arbitrary function, which is called the exponentially general convex function.
We investigate several properties of the exponentially general convex functions and discuss their
relations with convex functions. Optimality conditions are characterized by a class of variational
inequalities, which is called the exponentially general variational inequality. Several new
results characterizing the exponentially general convex functions are obtained. Results obtained in
this paper can be viewed as significant improvement of previously known results.

## A Finite Difference Method for Solution of Nonlinear Two Point Boundary Value Problem With a Neumann Boundary Conditions

### Author(s): PRAMOD KUMAR PANDEY and FAISAL AL-SHOWAIKH

**Abstract:** A finite difference scheme for the solution of two point boundary value problem in
ordinary differential equations subject to the Neumann boundary conditions presented in this article.
The propose scheme is tested on linear and non-linear problems. The solution of the discretized
problems was solved by iterative methods, i.e. Gauss-Seidel and Newton-Raphson method. The
computational results demonstrate reliability and efficiency of the developed finite difference method.
Moreover, numerical results confirm that scheme has second order accuracy.

## Opial Type Inequalities for Conformable Fractional Integrals via Convexity

### Author(s): MEHMET ZEKI SARIKAYA and CANDAN CAN BILISIK

**Abstract:** The main target addressed in this article are presenting Opial type inequalities
for Katugampola conformable fractional integral. In accordance with this purpose we try to use
more general type of function in order to make a generalization. Thus our results cover the
previous published studies for Opial type inequalities.

## On Some Common Fixed Points Theorems in Intuitionistic Fuzzy Metric Spaces and Its Applications

### Author(s): RAJINDER SHARMA and DEEPTI THAKUR

**Abstract:** In this paper, we established some common fixed point theorems for pairs of semi
compatible and occasionally weakly compatible mappings in an intuitionistic fuzzy metric space
(briefly IFM space) satisfying contractive type condition. In this paper, we observe that the notion
of common property (E.A.) relatively relaxes the required containment of the range of one mapping
into the range of other which is utilized to construct the sequence of joint iterates. We extended
the results established in [13] to intuitionistic fuzzy metric space.

## New Representations of Integrals of Polylogarithmic Functions With Quadratic Argument

### Author(s): A. SOFO and G. SORRENTINO

**Abstract:** In this paper we explore the representation and many connections between integrals
of products of polylogarithmic functions with a quadratic argument and Euler sums. The connection
between polylogarithmic functions and Euler sums is well known. Some examples of integrals of products
of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given.

## Voronec Equations for Nonlinear Nonholonomic Systems

### Author(s): FEDERICO TALAMUCCI

**Abstract:** One of the founders of the mechanics of nonoholonomic systems is Voronec who
published in 1901 a significant generalization of the Čaplygin’s equations, by removing some
restrictive assumptions. In the frame of nonholonomic systems, the Voronec equations are probably
less frequent and common with respect to the prevalent methods of quasi–coordinates
(Hamel–Boltzmann equations) and of the acceleration energy (Gibbs–Appell equations). In this paper
we start from the case of linear nonholonomic constraints, in order to extend the Voronec equations
to nonlinear nonholonomic systems. The comparison between two ways of expressing the equations of
motion is performed. We ﬁnally comment that the adopted procedure is appropriated to implement
further extensions.

## On Certain Class of Meromorphic Functions With Positive Coefficients Defined by Rapid Operator

### Author(s): B. VENKATESWARLU, P. THIRUPATHI REDDY and R. MAHDURI SHILPA

**Abstract:** The aim of the present paper is to introduce a class
${\sum}_{p}^{\ast}\left(A,B,\mu ,\mathit{\theta}\right)$ of meromorphic
univalent functions in $E=\left\{0<\left|z\right|<1\right\}$ and
investigate coefficient estimates, distortion properties and radius of convexity estimates for this
class. Furthermore, it is shown that the class ${\sum}_{p}^{\ast}\left(A,B,\mu ,\mathit{\theta}\right)$ is closed under convex linear combinations, convolutions and
integral transforms.