Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 160
Author(s):  
Likai Liu ◽  
Jin-Lin Liu
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Lemniscate Of Bernoulli

Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0<γ≤1). Notably, these results are applied to derive sufficient conditions for f∈A to satisfy the condition zf′(z)f(z)2−1<1. Several previous results are extended.

scholarly journals On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohsan Raza ◽  
Hira Naz ◽  
Sarfraz Nawaz Malik ◽  
Sahidul Islam
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

scholarly journals Differential Subordinations for Nonanalytic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Georgia Irina Oros ◽  
Gheorghe Oros
Keyword(s):  
Complex Plane ◽  
Classical Theory ◽  
Analytic Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Analytic Case ◽  
Complex Functions ◽  
Differential Subordinations

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classesC1(U), respectively, andC2(U)to be univalent and to mapUonto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the classC1which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classesC1andC2following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). LetΩbe any set in the complex planeC, letpbe a nonanalytic function in the unit discU,p∈C2(U),and letψ(r,s,t;z):C3×U→C. In this paper, we consider the problem of determining properties of the functionp, nonanalytic in the unit discU, such thatpsatisfies the differential subordinationψ(p(z),Dp(z),D2p(z)-Dp(z);z)⊂Ω⇒p(U)⊂Δ.

scholarly journals On applications of differential subordination and superordination

2008 ◽  
Vol 39 (2) ◽  
pp. 155-164
Author(s):  
N. Marikkannan ◽  
C. Ganesamoorthy
Keyword(s):  
Linear Operator ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Ruscheweyh Derivative ◽  
Differential Subordination And Superordination

In the present investigation we obtain the sufficient conditions for normalized analytic functions $f$ to satisfy$$ q_1 \prec \frac{f^2}{z^2f'} \prec q_2, $$where $ q_1 $ and $ q_2 $ are univalent functions with $ q_1(0)= q_2(0)=1 $. Also we obtain the sandwich results involving Carlson-Shaffer linear operator, $ S\u{a}l\u{a} $gean derivative and Ruscheweyh derivative.

scholarly journals Extension of Nunokawa Lemma for Functions with Fixed Second Coefficient and Its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mostafa Amani ◽  
Rasoul Aghalary ◽  
Ali Ebadian
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Differential Inequalities ◽  
Initial Coefficient ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

In this paper, we study some properties of analytic functions with fixed initial coefficients. The methodology of differential subordination is used for modification and improvements of several well-known results for subclasses of univalent functions by restricting the functions with fixed initial coefficients. Actually, by extending the Nunokawa lemma for fixed initial coefficient functions, we obtain some novel results on subclasses of univalent functions, such as differential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new sufficient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with fixed second coefficients.

scholarly journals Simple Sufficient Subordination Conditions for Close-to-Convexity

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 241 ◽  
Author(s):  
Ebrahim Analouei Adegani ◽  
Teodor Bulboacă ◽  
Ahmad Motamednezhad
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination

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 .

scholarly journals Sufficiency Criteria for q -Starlike Functions Associated with Cardioid

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Saira Zainab ◽  
Ayesha Shakeel ◽  
Muhammad Imran ◽  
Nazeer Muhammad ◽  
Hira Naz ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Image Domain ◽  
Lemniscate Of Bernoulli ◽  
Differential Subordinations

This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q   h z / h n   z   ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.

scholarly journals On application of differential subordination for Carathéodory functions

2020 ◽  
Vol 12 (2) ◽  
pp. 370-382
Author(s):  
V. Agnes Sagaya Judy Lavanya ◽  
M. P. Jeyaraman ◽  
H. Aaisha Farzana
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Carathéodory Functions ◽  
Special Cases

AbstractNew sufficient conditions involving the properties of analytic functions to belong to the class of Carathéodory functions are investigated. Certain univalence and starlikeness conditions are deduced as special cases of main results.

scholarly journals Starlikeness associated with lemniscate of Bernoulli

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1937-1955 ◽  
Author(s):  
Vibha Madaan ◽  
Ajay Kumar ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Second Order ◽  
Lemniscate Of Bernoulli

For an analytic function f on the unit disk D = {z : |z|<1} satisfying f (0) = 0 = f'(0) - 1, we obtain sufficient conditions so that f satisfies |(z f'(z)/f(z))2 - 1|< 1. The technique of differential subordination of first and second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.

Differential sandwich theorems of p-valent functions defined by a certain linear operator

2016 ◽  
Vol 53 (2) ◽  
pp. 131-137
Author(s):  
Ping He ◽  
Defei Zhang
Keyword(s):  
Linear Operator ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Sandwich Type ◽  
Differential Subordination And Superordination

In this paper we introduce differential subordination and superordination properties for certain subclasses of analytic functions involving certain linear operator, and obtain sandwich-type results for the functions belonging to these classes.

scholarly journals QKtype spaces of analytic functions

2006 ◽  
Vol 4 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Hasi Wulan ◽  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient Conditions ◽  
Type Space ◽  
Necessary And Sufficient ◽  
Spaces Of Analytic Functions

For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

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