<abstract><title style='display:none'>Abstract</title>By this research notes, the well-known Tremblay operator and certain core knowledge therewith are firstly introduced and an extensive result containing numerous (analytic and) geometric properties (of possible applications of the related operator) along with a number of special implications are then constituted. As method for proving, the well-known assertion proposed by [8] is also considered there.</abstract>

Geometric properties of some applications of the Tremblay operator

<abstract><title style='display:none'>Abstract</title>In this paper we consider two continuous functions <italic>f, g</italic> : [<italic>a, b</italic>] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point <italic>x</italic> = <italic>a</italic> and we calculate its derivative.</abstract>

Properties of the intermediate point from a mean value theorem of the integral calculus - II

<abstract><title style='display:none'>Abstract</title>In this paper, we find conditions under which the bracket defined by a graded derivation on a Lie superalgebra (<italic>g</italic>, [, ]) is skew-supersymmetry and satisfies the super Jacobi identity, so it defines the structure of a Lie superalgebra on <italic>g.</italic>In the case of the algebra of differential forms on a supermanifold, we study the graded commutator of graded derivations, graded skew-derivations and a graded derivation, with another graded skew-derivation of the superalgebra of differential forms on a supermanifold.</abstract>

On graded derivations of superalgebras

<abstract><title style='display:none'>Abstract</title>In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.</abstract>

A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative

<abstract><title style='display:none'>Abstract</title>Indulal and Balakrishnan (2016) have put forward the Indu-Bala product and based on this product four new operations are defined by the authors of this manuscript in the paper “Four new operations of graphs based on Indu-Bala product and the Zagreb indices”. In this paper we establish explicit formulas of the forgotten topological index and reduced second Zagreb index in connection with these new operations of graphs.</abstract>

Forgotten topological index and reduced Zagreb index of four new operations of graphs

<abstract><title style='display:none'>Abstract</title>Inequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel.</abstract>

Generalized Briot-Bouquet differential equation based on new differential operator with complex connections

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we establish new iterative methods for the solution of scalar equations by using the decomposition technique given in [<xref ref-type="bibr" rid="j_gm-2023-0009_ref_016">16</xref>].
</abstract>

Iterative methods for solving scalar equations

<abstract>
<title style='display:none'>Abstract</title>
In this paper, using the method of Hicks, we prove some new related fixed point theorems for pair of mappings on two <italic>d</italic>-complete topological spaces under the suitable conditions. We also derive some related fixed point theorems from our main results. Finally, by using <italic>w</italic>-continuity, we establish a related fixed point theorem in <italic>d</italic>-complete Hausdorff topological spaces.
</abstract>

On related fixed points in two <italic>d</italic>-complete topological spaces

<abstract>
<title style='display:none'>Abstract</title>
Let 𝒜 be the class of normalized functions which are analytic in the open unit disc. Jakson <italic>q</italic>-derivative represented by convolution operator
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2023-0011_eq_001.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mi>q</mml:mi></mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>z</mml:mi></mml:mfrac><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>★</mml:mo><mml:mfrac><mml:mi>z</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>q</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mi>z</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow> <mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝒜</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
<tex-math>{D_q}f\left( z \right) = {1 \over z}\left\{ {f\left( z \right)\star{z \over {\left( {1 - qz} \right)\left( {1z} \right)}}} \right\},\,\,\,\,f \in \mathcal{A},\,\,\,q \in \left( {0,1} \right)</tex-math>
</alternatives>
</disp-formula>
is used to introduce a unified class <italic>Mq</italic>(<italic>α, β, γ</italic>), <italic>α</italic> ≥ 0, <italic>β, γ</italic> ∈ [0, 1) and its various mapping properties are studied. For <italic>q</italic> → 1−1, <italic>β</italic> = <italic>γ</italic> = 0 and <italic>α</italic> = 1, this class reduces to the class <italic>K</italic> of close-to-convex univalent functions. A number of interesting results such as <italic>q</italic>-Bernardi integral operator and inclusion relations are included as part of this study. Applications are also pointed out as consequences of the main results.
</abstract>

On analytic functions related with <italic>q</italic>-generalized bounded Mocanu variation

<abstract>
<title style='display:none'>Abstract</title>
In this paper, in the first part, we will improve and extend the main results 1 and 2 of Mishra et al. [6] in a <italic>b</italic>-metric space, and in the second and last part, we will improve and extend Theorem 5 and Theorem 6 of Shah [7] in a <italic>b</italic>-metric-like space, in both sections, we will use our new notion of occasionally weakly biased maps of type (𝒜). Some illustrative examples are given in order to clarify and support our findings.
</abstract>

Unique common fixed point results for two maps in incomplete spaces

<abstract>
<title style='display:none'>Abstract</title>
This work studies a mathematical model involving a dynamic contact between two elasto-viscoplastic piezoelectric bodies with damage. The contact is modeled on a combination of a normal compliance and a normal damped response law associated with friction. We derive a variational formulation of the problem and we prove the existence and uniqueness result of the weak solution. The proof is based on the classical existence and uniqueness result for parabolic inequalities, differential equations and fixed-point arguments.
This paper examines a friction problem between two piezoelectric bodies that are elastic- viscoplastic. The study uses both numerical and analytical methods. The contact surface is assumed to have a thin layer of lubricant, and a damped response contact condition is considered. The problem also takes into account electrical and frictional effects. The paper proves the existence of a unique weak solution to the problem by using variational inequalities and a fixed point argument. Additionally, a fully discrete approximation is investigated using the Euler scheme and finite element method for the spatial variable and time derivatives. Error estimates are then calculated, leading to convergence results under certain regularity conditions.
</abstract>

Variational and numerical analysis of a dynamic elasto-viscoplastic piezoelectric frictional contact problem

<abstract>
<title style='display:none'>Abstract</title>
We investigate random variables for which the variance and the information potential satisfy a preservation law.
</abstract>

Variance and information potential of some random variables

<abstract><title style='display:none'>Abstract</title>In this scientific note, an operator, which is the well-known Tremblay operator in the literature, is first introduced and some of its applications to certain analytic complex functions, which are normalized and analytic in the open unit disk, are then determined. In addition, certain special results of the related applications are also emphasized.</abstract>

Certain basic information related to the Tremblay operator and some applications in connection therewith

<abstract><title style='display:none'>Abstract</title>In this study, the author has established a new lemma and Bullen-type inequalities for conformable fractional integrals. Also, it is given some applications involving Bullen type integral inequalities for differentiable functions to show the results.</abstract>

Some Bullen-type inequalities for conformable fractional integrals

<abstract><title style='display:none'>Abstract</title>In the present investigation, by making use of strong differential subordinations and superordinations, we introduce and study two new classes of holomorphic functions containing generalized differential operator. Also we determine important properties for functions belongs to these classes.</abstract>

Two New Classes of Analytic Functions Defined by Strong Differential Subordinations and Superordinations

<abstract><title style='display:none'>Abstract</title>We consider classical entropies associated with several continuous distributions of probabilities. Explicit expressions and properties of them are presented.</abstract>

Entropies related to integral operators

Open Access License