# Volume 12 (2020), Number 2

## New Differential Operator Involving the $q$-Ruscheweyh Derivative and the Symmetric Differential

### Author(s): KHALID ALSHAMMARI and MASLINA DARUS

**Abstract:** In this paper, we define a new differential operator involving
$q$-calculus
by using a familiar technique which is a Hadamard product. We use convolution between
$q$-analogue
of the Ruscheweyh derivative by Aldweby and Darus [1] and the symmetric Salagean differential
operator by Ibrahim and Darus [2]. We study some properties of the new operator such as the
negative coefficients estimate, growth and the radii of starlikeness and convexity.

## A Landau’s Inequality for Semigroups Revisited

### Author(s): GEORGE A. ANASTASSIOU

**Abstract:** H. Kraljevič and S. Kurepa in 1970 ([13]) proved a nice Landau type
inequality for semigroups by making use of Taylor’s formula with integral remainder for
semigroups, which has been a traditional method for proving such inequalities. The author
offers another method in the semigroups and reproves the above result by using his earlier
result about Ostrowski inequalities for semigroups, see [1], Chapter 16, pp. 259-289. He finds
his method easier and convenient. An application is given at the end.

## Positive Solutions to Caputo-Hadamard Fractional Integro-Differential Equations With Integral Boundary Conditions

### Author(s): ABDELOUAHEB ARDJOUNI

**Abstract:** In this article, we study the existence and uniqueness of positive
solutions of a Caputo-Hadamard fractional integro-differential equation with integral boundary
conditions. The fixed point theorems and the method of upper and lower solutions are used to
obtain the desired results. An example illustrating the main results is presented.

## An Integral Equation From Physics - a Synthesis Survey - Part III

### Author(s): MARIA DOBRIȚOIU

**Abstract:** This is the third part of the synthesis survey on the study of the integral
equation from physics:
$$x\left(t\right)={\displaystyle \underset{a}{\overset{b}{\int}}K\left(t,s,x\left(s\right),x\left(a\right),x\left(b\right)\right)ds}+f\left(t\right),\text{\hspace{0.17em}}t\in \left[a,b\right],$$
and contains the results concerning the approximation of its solution by applying numerical
methods. This part ends with an example.

## Some Integral Inequalities for Operator Monotone Functions

### Author(s): SILVESTRU SEVER DRAGOMIR

**Abstract:** In this paper we prove among others that
$$\underset{T}{\int}{p}_{t}f\left({A}_{t}\right){A}_{t}d\mu}\left(t\right)\u2a7e{\displaystyle \underset{T}{\int}{p}_{s}f\left({A}_{s}\right)d\mu}\left(s\right){\displaystyle \underset{T}{\int}{p}_{t}{A}_{t}d\mu \left(t\right)$$
for an operator monotone function $f$ on $\left(\mathrm{0,}\infty \right)$, where ${\left({A}_{t}\right)}_{t\in T}$ is a bounded continuous field of
positive operators in $\mathfrak{B}\left(H\right)$ defined on a locally compact Hausdorff space $f$ with a bounded Radon measure
$\mu $ that satisfy certain conditions while
${\left({p}_{t}\right)}_{t\in T}$ are nonnegative with ${\int}_{T}{p}_{t}d\mu}\left(t\right)=1$. Some particular inequalities of interest are also provided.

## Characterizations of Strongly Generalized Convex Functions

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

**Abstract:** In this paper, we define and consider some new concepts of the strongly
general convex functions with respect to two arbitrary functions. Some properties of the
strongly generalized convex functions are investigated under suitable conditions. It is shown
that the optimality conditions of the strongly generalized convex functions are characterized
by a class of variational inequalities, which is called the strongly generalized variational
inequality. Some special cases also discussed. Results obtained in this paper can be viewed as
refinement and improvement of previously known results.

## On the Generalized Ostrowski Type Inequalities via Tempered Fractional Integrals

### Author(s): MEHMET ZEKI SARIKAYA and MEHMET EYÜP Kİ̇Rİ̇Ş

**Abstract:** In this paper, we have found a new version of the Ostrowski type inequality
for tempered Riemann-Liouville fractional integrals. In addition, some relevant results have
been obtained.

## Math Problems - Solutions, Observations and Comments

### Author(s): MARIA DOBRIȚOIU

**Abstract:** The paper aims to present some theoretical methodological considerations
regarding the solving of math problems. Thus, there are presented theoretical aspects regarding
the formation of mathematical thought, the activity of education and training through
mathematics, types of learning in mathematics and forms of organizing the activities in
mathematical education in Romania, reason for which the References that were used are entirely
in Romanian. The paper also contains some math problems solved: geometry, algebra or even
arithmetic problems, and the solutions are accompanied by observations and possible methodical
comments.