New Class of Specific Functions with Fractional Derivatives

Güney, Hatun Özlem; Owa, Shigeyoshi

doi:10.3390/axioms14080608

Open AccessFeature PaperArticle

New Class of Specific Functions with Fractional Derivatives

by

Hatun Özlem Güney

¹

and

Shigeyoshi Owa

^2,*

¹

Department of Mathematics, Faculty of Science, Dicle University, 21280 Diyarbakır, Turkey

²

“1 Decembrie 1918” University Alba Iulia, 510009 Alba-Iulia, Romania

^*

Author to whom correspondence should be addressed.

Axioms 2025, 14(8), 608; https://doi.org/10.3390/axioms14080608

Submission received: 25 June 2025 / Revised: 31 July 2025 / Accepted: 31 July 2025 / Published: 5 August 2025

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Related Topics)

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Abstract

Let

A_{n}

be the class of specific analytic functions

f (z) = z + \sum_{k = 1}^{\infty} a_{1 + \frac{k}{n}} z^{1 + \frac{k}{n}} (n \in N = {1, 2, 3, \dots})

in the open unit disk

U .

For

f (z) \in A_{n},

fractional derivatives

D_{z}^{λ} f (z)

and

D_{z}^{j + λ} f (z)

(0 \leq λ < 1, j \in N)

are defined by using Gamma functions. Applying such fractional derivatives, we introduce a new subclass

A_{n} (j, λ, α, β)

of

A_{n} .

In this paper, we establish sufficient conditions for

f (z)

for

A_{n} (j, λ, α, β),

coefficient inequalities for

| a_{1 + \frac{1}{n}} |

and

| a_{1 + \frac{k}{n}} |

(k = 2, 3, 4, \dots)

of

f (z) \in A_{n} (j, λ, α, β),

and some interesting argument properties of fractional derivatives for

f (z) \in A_{n}

through an example.

Keywords: specific function; fractional integral; fractional derivative; subordination; coefficient inequality; argument property

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MDPI and ACS Style

Güney, H.Ö.; Owa, S. New Class of Specific Functions with Fractional Derivatives. Axioms 2025, 14, 608. https://doi.org/10.3390/axioms14080608

AMA Style

Güney HÖ, Owa S. New Class of Specific Functions with Fractional Derivatives. Axioms. 2025; 14(8):608. https://doi.org/10.3390/axioms14080608

Chicago/Turabian Style

Güney, Hatun Özlem, and Shigeyoshi Owa. 2025. "New Class of Specific Functions with Fractional Derivatives" Axioms 14, no. 8: 608. https://doi.org/10.3390/axioms14080608

APA Style

Güney, H. Ö., & Owa, S. (2025). New Class of Specific Functions with Fractional Derivatives. Axioms, 14(8), 608. https://doi.org/10.3390/axioms14080608

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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New Class of Specific Functions with Fractional Derivatives

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