scholarly journals q-Analogue of Differential Subordinations

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 724 ◽  
Author(s):  
Meraj Ul-Haq ◽  
Mohsan Raza ◽  
Muhammad Arif ◽  
Qaiser Khan ◽  
Huo Tang
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Janowski Functions ◽  
Differential Subordinations

In this article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on α such that 1 + α z ∂ q h z h z n ( n = 0 , 1 , 2 , 3 ) are subordinated by Janowski functions and h z ≺ 1 + 4 3 z + 2 3 z 2 . We also consider the same implications such that h z ≺ 1 + 2 z + 1 2 z 2 . We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon.

Sufficient conditions for p-valent functions

2021 ◽  
Vol 71 (5) ◽  
pp. 1089-1102
Author(s):  
Qaiser Khan ◽  
Jacek Dziok ◽  
Mohsan Raza ◽  
Muhammad Arif
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Exponential Functions ◽  
Janowski Functions ◽  
Current Article

Abstract In the current article, we examine some properties of analytic functions associated with cosine and exponential functions. We calculate some conditions on α so that; if 1 + α z 2 − p f ′ ( z ) p $1+\frac{\alpha z^{2-p}f'(z)}{p}$ , 1 + α z 2 f ′ ( z ) p f ( z ) $1+\alpha \frac{z^{2}f'(z)}{pf(z)}$ , 1 + α z p + 2 f ′ ( z ) p f 2 ( z ) $1+\alpha \frac{z^{p+2}f'(z)}{pf^{2}(z)}$ and 1 + α z 2 p + 2 f ′ ( z ) p f 3 ( z ) $1+\alpha \frac{z^{2p+2}f'(z)}{pf^{3}(z)}$ are subordinated by Janowski functions, then f ( z ) z p ≺ cos ⁡ ( z ) $\frac{f(z)}{z^{p}}\prec \cos (z)$ . Further, we also discuss the same implications for f ( z ) z p ≺ e z $\frac{f(z)}{z^{p}}\prec \textrm{e}^{z}$ .

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 A Subclass of Janowski Starlike Functions Involving Mathieu-Type Series

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 2
Author(s):  
Dong Liu ◽  
Serkan Araci ◽  
Bilal Khan
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Function Class ◽  
Partial Sums ◽  
Invariant Functions ◽  
Type Series ◽  
Janowski Functions ◽  
Radius Problems ◽  
Symmetrical Points

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived.

scholarly journals Differential subordination for Janowski functions with positive real part

2021 ◽  
Vol 66 (3) ◽  
pp. 457-470
Author(s):  
Swati Anand ◽  
V. Ravichandran ◽  
Sushil Kumar
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions ◽  
Differential Subordination ◽  
Positive Real Part ◽  
Algebraic Condition ◽  
Positive Real ◽  
First Order ◽  
Janowski Functions ◽  
Subordination Theory ◽  
Differential Subordinations

"Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition.We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain suffcient conditions for normalized analytic functions to be Janowski starlike functions."

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.

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 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.

scholarly journals Poisson distribution series on a general class of analytic functions

2020 ◽  
Vol 24 (2) ◽  
pp. 241-251
Author(s):  
Basem A. Frasin
Keyword(s):  
Integral Operator ◽  
Poisson Distribution ◽  
Analytic Functions ◽  
General Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Class A ◽  
Poisson Distribution Series ◽  
Necessary And Sufficient

The main object of this paper is to find necessary and sufficient conditions for the Poisson distribution series to be in a general class of analytic functions with negative coefficients. Further, we consider an integral operator related to the Poisson distribution series to be in this class. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

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