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10 pages, 267 KiB

Open AccessArticle

Strong Sandwich-Type Results for Fractional Integral of the Extended q-Analogue of Multiplier Transformation

by Alina Alb Lupaş

Mathematics 2024, 12(18), 2830; https://doi.org/10.3390/math12182830 - 12 Sep 2024

Viewed by 647

In this research, we obtained several strong differential subordinations and strong differential superordinations, which gave sandwich-type results for the fractional integral of the extended q-analogue of multiplier transformation. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

15 pages, 322 KiB

Open AccessFeature PaperArticle

Strong Differential Subordinations and Superordinations for Riemann–Liouville Fractional Integral of Extended q-Hypergeometric Function

by Alina Alb Lupaş and Georgia Irina Oros

Mathematics 2023, 11(21), 4474; https://doi.org/10.3390/math11214474 - 28 Oct 2023

Cited by 4 | Viewed by 1332

Abstract

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory)

19 pages, 324 KiB

Open AccessArticle

New Results on a Fractional Integral of Extended Dziok–Srivastava Operator Regarding Strong Subordinations and Superordinations

by Alina Alb Lupaş

Symmetry 2023, 15(8), 1544; https://doi.org/10.3390/sym15081544 - 5 Aug 2023

Cited by 2 | Viewed by 1088

Abstract

In 2012, new classes of analytic functions on

U \times \bar{U}

with coefficient holomorphic functions in

\bar{U}

were defined to give a new approach to the concepts of strong differential subordination and strong differential superordination. Using those new classes, the extended [...] Read more.

In 2012, new classes of analytic functions on

U \times \bar{U}

with coefficient holomorphic functions in

\bar{U}

were defined to give a new approach to the concepts of strong differential subordination and strong differential superordination. Using those new classes, the extended Dziok–Srivastava operator is introduced in this paper and, applying fractional integral to the extended Dziok–Srivastava operator, we obtain a new operator

D_{z}^{- γ} H_{m}^{l} [α_{1}, β_{1}]

that was not previously studied using the new approach on strong differential subordinations and superordinations. In the present article, the fractional integral applied to the extended Dziok–Srivastava operator is investigated by applying means of strong differential subordination and superordination theory using the same new classes of analytic functions on

U \times \bar{U} .

Several strong differential subordinations and superordinations concerning the operator

D_{z}^{- γ} H_{m}^{l} [α_{1}, β_{1}]

are established, and the best dominant and best subordinant are given for each strong differential subordination and strong differential superordination, respectively. This operator may have symmetric or asymmetric properties. Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

16 pages, 321 KiB

Open AccessArticle

Strong Differential Subordination and Superordination Results for Extended q-Analogue of Multiplier Transformation

by Alina Alb Lupaş and Firas Ghanim

Symmetry 2023, 15(3), 713; https://doi.org/10.3390/sym15030713 - 13 Mar 2023

Cited by 6 | Viewed by 1639

Abstract

The results obtained by the authors in the present article refer to quantum calculus applications regarding the theories of strong differential subordination and superordination. The q-analogue of the multiplier transformation is extended, in order to be applied on the specific classes of functions involved in strong differential subordination and superordination theories. Using this extended q-analogue of the multiplier transformation, a new class of analytic normalized functions is introduced and investigated. The convexity of the set of functions belonging to this class is proven and the symmetry properties derive from this characteristic of the class. Additionally, due to the convexity of the functions contained in this class, interesting strong differential subordination results are proven using the extended q-analogue of the multiplier transformation. Furthermore, strong differential superordination theory is applied to the extended q-analogue of the multiplier transformation for obtaining strong differential superordinations for which the best subordinants are provided. Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

13 pages, 484 KiB

Open AccessArticle

Applications of Confluent Hypergeometric Function in Strong Superordination Theory

by Georgia Irina Oros, Gheorghe Oros and Ancuța Maria Rus

Axioms 2022, 11(5), 209; https://doi.org/10.3390/axioms11050209 - 29 Apr 2022

Cited by 6 | Viewed by 2691

Abstract

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory)

12 pages, 791 KiB

Open AccessArticle

Strong Differential Superordination Results Involving Extended Sălăgean and Ruscheweyh Operators

by Alina Alb Lupaş and Georgia Irina Oros

Mathematics 2021, 9(19), 2487; https://doi.org/10.3390/math9192487 - 4 Oct 2021

Cited by 6 | Viewed by 1427

Abstract

The notion of strong differential subordination was introduced in 1994 and the theory related to it was developed in 2009. The dual notion of strong differential superordination was also introduced in 2009. In a paper published in 2012, the notion of strong differential [...] Read more.

U \times \bar{U}

having as coefficients holomorphic functions in

\bar{U}

. Using those new classes, extended Sălăgean and Ruscheweyh operators were introduced and a new extended operator was defined as

L_{α}^{m} : A_{n ζ}^{*} \to A_{n ζ}^{*},

L_{α}^{m} f (z, ζ) = (1 - α) R^{m} f (z, ζ) + α S^{m} f (z, ζ),

z \in U,

ζ \in \bar{U},

where

R^{m} f (z, ζ)

is the extended Ruscheweyh derivative,

S^{m} f (z, ζ)

is the extended Sălăgean operator and

A_{n ζ}^{*} = {f \in H (U \times \bar{U}), f (z, ζ) = z + a_{n + 1} (ζ) z^{n + 1} + \dots, z \in U,

ζ \in \bar{U}} .

This operator was previously studied using the new approach on strong differential subordinations. In the present paper, the operator is studied by applying means of strong differential superordination theory using the same new classes of analytic functions on

U \times \bar{U} .

Several strong differential superordinations concerning the operator

L_{α}^{m}

are established and the best subordinant is given for each strong differential superordination. Full article

(This article belongs to the Special Issue Complex Analysis and Its Applications)

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