<abstract xmlns="http://www.w3.org/1999/xhtml">

In this investigation, we introduce a new <italic>q</italic>-differential operator that generalizes the well-known Opoola differential operator. Using the operator, we define a family of <italic>q</italic>-bounded turning functions. The new class is denoted by ℛ<italic>qn</italic> (<italic>µ, β, τ </italic>; <italic>λ</italic>). Afterwards, inclusion, characterisation, growth, distortion, covering, linear combination and neighborhood properties of functions <italic>f ∈ </italic>ℛ<italic>qn</italic> (<italic>µ, β, τ </italic>; <italic>λ</italic>) are presented.
</abstract>

Faculty of Physical Sciences Department of Mathematics

Some Properties of a Family of Univalent Functions Defined by a Generalized Opoola Differential Operator

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, some common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings satisfying integral type are obtained. Our results improve several results especially the results of [<xref ref-type="bibr" rid="j_gm-2022-0002_ref_003">3</xref>], Pathak et al. [<xref ref-type="bibr" rid="j_gm-2022-0002_ref_013">13</xref>], Djoudi and Aliouche [<xref ref-type="bibr" rid="j_gm-2022-0002_ref_006">6</xref>], Mbarki [<xref ref-type="bibr" rid="j_gm-2022-0002_ref_010">10</xref>] and others.
</abstract>

Coincidence and Common Fixed Points of Infinite Family of Mappings under Weaker Conditions

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we introduce a new class of generalized convex functions called (<italic>h, r</italic>)-preinvex functions, and establish some new Hermite-Hadamard type inequalities under (<italic>h, r</italic>)-preinvexity.
</abstract>

Laboratoire des télécommunications Faculté des Sciences

Laboratory of advanced materials.University of Badji Mokhtar of Annaba Faculty of sciences

Hermite-Hadamard’s inequalities for (<italic>h, r</italic>)-preinvex functions

<abstract xmlns="http://www.w3.org/1999/xhtml">

The aim of this paper is to generalize the convexity preserving properties of sets by direct and inverse images initially stated for set-valued ratios of affine functions to a broader class of fractional type set-valued functions which strictly includes the first one.
</abstract>

An extension of the class of set-valued ratios of affine functions

<abstract xmlns="http://www.w3.org/1999/xhtml">

This paper consists of Hankel results of <italic>ω</italic>-order reversing partial contraction as a semigroup of linear operator. Spectral mapping theorem was investigated and it was established that <italic>ω</italic>-order reversing partial contraction mapping is an eventually uniform continuous semigroup in which we showed that the proof of the extension of <italic>T </italic>(<italic>t</italic>) to a <italic>C</italic>0-semigroup ̂<italic>T</italic>(<italic>t</italic>) on a space ̂<italic>X </italic>containing <italic>X </italic>isometrically bounded. The space is constructed in such a way that the spectrum of the generator <italic>A </italic>of <italic>T </italic>(<italic>t</italic>) coincides with the spectrum of the generator <italic>Â </italic>of ̂<italic>T</italic>(<italic>t</italic>) and the approximate point spectrum of <italic>A </italic>coincides with the point spectrum of <italic>Â</italic>.
</abstract>

Morgantown Faculty of Physical Sciences Department of Mathematics

Faculty of Physical Sciences, Department of Mathematics

Results of Hankel Semigroup of Linear Operator as Eventually Uniform Continuous Semigroup

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, the authors established a new identity for differentiable functions, afterward, they obtained some new general inequalities for functions whose first derivatives in absolute value at certain powers are <italic>h</italic>-convex by using the identity. On the other hand, a general inequality is studied, which gives Hermite-Hadamard, Bullen and Simpson inequalities. Also, they gave some applications for special means for arbitrary positive numbers.
</abstract>

Mustafa Kemal University, Yayladağı Vocational School

Department of Tax and Accounting, Mustafa Kemal University

Mustafa Kemal University, Faculty of Science

Generalizations of Hermite-Hadamard, Bullen and Simpson inequalities via <italic>h−</italic>convexity

<abstract xmlns="http://www.w3.org/1999/xhtml">

For a continuous and positive function <italic>w </italic>(<italic>λ</italic>), <italic>λ > </italic>0 and <italic>µ </italic>a positive measure on (0, <italic><bold>∞</bold></italic>) we consider the following <italic>integral transform</italic>
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></mrow></mstyle><mo>,</mo></mrow></math>
<tex-math>\mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \right)} ,</tex-math>
</alternatives>
</disp-formula>
where the integral is assumed to exist for <italic>T </italic>a positive operator on a complex Hilbert space <italic>H</italic>.
We show among others that, if <italic>B, A > </italic>0, then
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>−</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mrow><mo>(</mo><mrow><mi>B</mi><mo>−</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>λ</mi><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>t</mi></mrow><mo>)</mo></mrow><mi>B</mi><mo>+</mo><mi>t</mi><mi>A</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mi>B</mi><mo>−</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mi>d</mi><mi>t</mi></mrow></mrow></mstyle></mrow><mo>)</mo></mrow><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>.</mo></mrow></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{{\left[{\mathcal{D}\left({w,\mu}\right)\left(A\right)-\mathcal{D}\left({w,\mu}\right)\left(B\right)}\right]\left({B-A}\right)}\cr{=\int_0^\infty{w\left(\lambda\right)\left({\int_0^1{{{\left[{\lambda+\left({1-t}\right)B+tA{)^{-1}}\left({B-A}\right)}\right]}^2}dt}}\right)d\mu\left(\lambda\right).}}\cr}</tex-math>
</alternatives>
</disp-formula>
We also provide some sufficient conditions for the operators <italic>A, B > </italic>0 such that the inequality
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2022-0007_eq_003.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>B</mi><mo>+</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mi>A</mi><mo>≥</mo><mi>A</mi><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mi>B</mi><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></math>
<tex-math>\mathcal{D}\left({w,\mu}\right)\left(A\right)B+\mathcal{D}\left({w,\mu}\right)\left(B\right)A{\ge}A\mathcal{D}\left({w,\mu}\right)\left(A\right)+B\mathcal{D}\left({w,\mu}\right)\left(B\right)</tex-math>
</alternatives>
</disp-formula>
holds. Some examples for power and logarithmic functions are also provided.
</abstract>

Simple Operator Asynchronicity of an Integral Transform with Applications

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, unique common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings are obtained. This unique common fixed point is guaranteed under a new concept named a weak altering distance function. Our theorems improve some results especially the ones of [<xref ref-type="bibr" rid="j_gm-2022-0008_ref_002">2</xref>].
</abstract>

Badji Mokhtar-Annaba University Faculty of Sciences

Department of Mathematics Laboratory of Applied Mathematics

A Unique Common Fixed Point under a Weak Altering Distance Function

<abstract xmlns="http://www.w3.org/1999/xhtml">

We introduce a new iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued quasi-nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued quasi-nonexpansive. Futhermore, we applied our main result for finding a common solution of convex minimization problem and fixed points problem. Essentially a new approach for finding solutions of equilibrium problems and fixed point problems whith set-valued operators is provided.
</abstract>

A new fixed point technique for equilibrium problem with a finite family of multivalued quasi-nonexpansive mappings

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General Mathematics

Volume 30 (2022): Issue 1 (June 2022)

Some Properties of a Family of Univalent Functions Defined by a Generalized Opoola Differential Operator

Coincidence and Common Fixed Points of Infinite Family of Mappings under Weaker Conditions

Hermite-Hadamard’s inequalities for (h, r)-preinvex functions

An extension of the class of set-valued ratios of affine functions

Results of Hankel Semigroup of Linear Operator as Eventually Uniform Continuous Semigroup

Generalizations of Hermite-Hadamard, Bullen and Simpson inequalities via h−convexity

Simple Operator Asynchronicity of an Integral Transform with Applications

A Unique Common Fixed Point under a Weak Altering Distance Function

A new fixed point technique for equilibrium problem with a finite family of multivalued quasi-nonexpansive mappings

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