Research

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://0-doi-org.brum.beds.ac.uk/10.3390/math9192487 - 04 Oct 2021

Cited by 5 | Viewed by 1015

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.

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 subordination was given a new approach by defining new classes of analytic functions on

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)

14 pages, 318 KiB

Open AccessArticle

Bohr Radius Problems for Some Classes of Analytic Functions Using Quantum Calculus Approach

by Om Ahuja, Swati Anand and Naveen Kumar Jain

Mathematics 2020, 8(4), 623; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040623 - 18 Apr 2020

Cited by 5 | Viewed by 2480

Abstract

The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order

α

. The Bohr radius is also determined for a generalized class of q [...] Read more.

The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order

α

. The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients. Full article

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

14 pages, 298 KiB

Open AccessArticle

On Coefficient Functionals for Functions with Coefficients Bounded by 1

by Paweł Zaprawa, Anna Futa and Magdalena Jastrzębska

Mathematics 2020, 8(4), 491; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040491 - 01 Apr 2020

Cited by 2 | Viewed by 1605

Abstract

In this paper, we discuss two well-known coefficient functionals

a_{2} a_{4} - {a_{3}}^{2}

and

a_{4} - a_{2} a_{3}

. The first one is called the Hankel determinant of order 2. The second one is a special [...] Read more.

In this paper, we discuss two well-known coefficient functionals

a_{2} a_{4} - {a_{3}}^{2}

and

a_{4} - a_{2} a_{3}

. The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class

Q_{R} (\frac{1}{2})

of analytic functions with real coefficients which satisfy the condition

Re f (z) z > \frac{1}{2}

for z in the unit disk

Δ

. It is known that all coefficients of

f \in Q_{R} (\frac{1}{2})

are bounded by 1. We find the upper bound of

a_{2} a_{4} - {a_{3}}^{2}

and the bound of

| a_{4} - a_{2} a_{3} |

. We also consider a few subclasses of

Q_{R} (\frac{1}{2})

and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients. Full article

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

8 pages, 260 KiB

Open AccessFeature PaperArticle

Gamma-Bazilevič Functions

by Sa’adatul Fitri and Derek K. Thomas

Mathematics 2020, 8(2), 175; https://0-doi-org.brum.beds.ac.uk/10.3390/math8020175 - 02 Feb 2020

Cited by 2 | Viewed by 1720

Abstract

For

γ \geq 0

and

α \geq 0

, we introduce the class

B_{1}^{γ} (α)

of Gamma–Bazilevič functions defined for

z \in D

by [...] Read more.

For

γ \geq 0

and

α \geq 0

, we introduce the class

B_{1}^{γ} (α)

of Gamma–Bazilevič functions defined for

z \in D

by

R e \{{[\frac{z f^{'} (z)}{f {(z)}^{1 - α} z^{α}} + \frac{z f^{″} (z)}{f^{'} (z)} + (α - 1) (\frac{z f^{'} (z)}{f (z)} - 1)]}^{γ} {[\frac{z f^{'} (z)}{f {(z)}^{1 - α} z^{α}}]}^{1 - γ}\} > 0

. We shown that

B_{1}^{γ} (α)

is a subset of

B_{1} (α)

, the class of

B_{1} (α)

Bazilevič functions, and is therefore univalent in

D

. Various coefficient problems for functions in

B_{1}^{γ} (α)

are also given. Full article

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

12 pages, 264 KiB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Bi-Univalent Functions Defined by Using Differential Subordination and a Certain Fractional Derivative Operator

by Hari M. Srivastava, Ahmad Motamednezhad and Ebrahim Analouei Adegani

Mathematics 2020, 8(2), 172; https://0-doi-org.brum.beds.ac.uk/10.3390/math8020172 - 01 Feb 2020

Cited by 56 | Viewed by 2428

Abstract

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general [...] Read more.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients

| a_{n} |

of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results. Full article

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

12 pages, 257 KiB

Open AccessArticle

Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

by Kyung-Won Hwang and Cheon Seoung Ryoo

Mathematics 2019, 7(12), 1205; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121205 - 09 Dec 2019

Cited by 4 | Viewed by 1491

Abstract

The main goal of this paper is to study some interesting identities for the multiple twisted

(p, q)

-L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted [...] Read more.

The main goal of this paper is to study some interesting identities for the multiple twisted

(p, q)

-L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted

(p, q)

-Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted

(p, q)

-Euler zeta function and multiple twisted

(p, q)

-L-function, which interpolate the Carlitz-type higher order twisted

(p, q)

-Euler numbers and Carlitz-type higher order twisted

(p, q)

-Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted

(p, q)

-Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted

(p, q)

-L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted

(p, q)

-Euler numbers and polynomials by using the symmetric property for the multiple twisted

(p, q)

-L-function. Full article

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

14 pages, 286 KiB

Open AccessFeature PaperArticle

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

by Oh Sang Kwon and Young Jae Sim

Mathematics 2019, 7(8), 721; https://0-doi-org.brum.beds.ac.uk/10.3390/math7080721 - 08 Aug 2019

Cited by 11 | Viewed by 2254

Abstract

Let

{SR}^{*}

be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition

f (0) = 0 = f^{'} (0) - 1

, [...] Read more.

Let

{SR}^{*}

be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition

f (0) = 0 = f^{'} (0) - 1

,

Re {z f^{'} (z) / f (z)} > 0

, for

z \in D : = {z \in C : | z | < 1}

and

a_{n} : = f^{(n)} (0) / n!

is real for all

n \in N

. In the present paper, it is obtained that the sharp inequalities

- 4 / 9 \leq H_{3, 1} (f) \leq \sqrt{3} / 9

hold for

f \in {SR}^{*}

, where

H_{3, 1} (f)

is the third Hankel determinant of order 3 defined by

H_{3, 1} (f) = a_{3} (a_{2} a_{4} - a_{3}^{2}) - a_{4} (a_{4} - a_{2} a_{3}) + a_{5} (a_{3} - a_{2}^{2})

. Full article

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

13 pages, 260 KiB

Open AccessArticle

A New Subclass of Analytic Functions Defined by Using Salagean q-Differential Operator

by Muhammad Naeem, Saqib Hussain, Tahir Mahmood, Shahid Khan and Maslina Darus

Mathematics 2019, 7(5), 458; https://0-doi-org.brum.beds.ac.uk/10.3390/math7050458 - 21 May 2019

Cited by 19 | Viewed by 3297

Abstract

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these [...] Read more.

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied. Full article

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

9 pages, 236 KiB

Open AccessArticle

Simple Sufficient Subordination Conditions for Close-to-Convexity

by Ebrahim Analouei Adegani, Teodor Bulboacă and Ahmad Motamednezhad

Mathematics 2019, 7(3), 241; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030241 - 07 Mar 2019

Cited by 6 | Viewed by 2078

Abstract

Using several applications of the theory of differential subordination we obtain sufficient conditions for usually normalized analytic functions to belong to certain subclasses of close-to-convex functions and close-to-convex functions of order

α

. Full article

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

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