A New Class of Analytic Functions Associated with Sălăgean Operator

Abstract

The main object of the present paper is to investigate a number of useful properties such as inclusion relation, distortion bounds, coefficient estimates, and subordination results for a new subclass of analytic functions which are defined here by means of a linear operator. Several known consequences of the results are also pointed out.

Majorization Results for Certain Subfamilies of Analytic Functions

“…In recent years, several subfamilies of the set A were studied as a special case of the class S * ðΛÞ. For example, (i) if we take ΛðzÞ = 1 + Mz/1 + Nz with −1 ≤ N < M ≤ 1, then the deduced family S * ½M, N ≡ S * ð1 + M z/1 + NzÞ is described by the functions of the Janowski star-like family established in [16] and later studied in different directions in [17,18] (ii) the family S * L ≡ S * ðΛðzÞÞ with ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p was developed in [19] by Sokól and Stankiewicz. The image of the function ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p demonstrates that the image domain is bounded by the Bernoullis lemniscate right-half plan specified by jw 2 ðzÞ − 1j < 1 (iii) by selecting ΛðzÞ = 1 + sin z, the class S * ðΛðzÞÞ leads to the family S * sin which was explored in [20] while S * e ≡ S * ðe z Þ has been produced in the article [21] and later studied in [22] (iv) the family S * R ≡ S * ðΛðzÞÞ with ΛðzÞ = 1 + z/JðJ + zÞ/ðJ − zÞ,J = ffiffi ffi 2 p + 1 is studied in [23] while S * cos ≔ S * ðcos ðzÞÞ and S * cosh ≔ S * ðcosh ðzÞÞ were recently examined by Raza and Bano [24], and Abdullah et.al [25], respectively Now, let us take the nonvanishing analytic functions h 1 ðzÞ and h 2 ðzÞ in U d with h 1 ð0Þ = h 2 ð0Þ = 1: Then, the families defined in this article consist of functions f ðzÞ ∈ A whose ratios f ðzÞ/zqðzÞ and qðzÞ are subordinated to h 1 ðzÞ and h 2 ðzÞ, respectively, for some analytic function qðzÞ with qð0Þ = 1 as…”

Sharp Bounds of the Coefficient Results for the Family of Bounded Turning Functions Associated with a Petal-Shaped Domain

“…is described by the functions of the Janowski starlike family established in [1] and later studied in different directions in [2,3] (ii) The family S * L ≡ S * ðΛðzÞÞ with ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p was developed in [4] by Sokól and Stankiewicz. The image of the function ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p demonstrates that the image domain is bounded by the Bernoulli's lemniscate right-half plan specified by |w 2 − 1 | <1 (iii) By selecting ΛðzÞ = 1 + sin z, the class S * ðΛðzÞÞ lead to the family S * sin which was explored in [5] while S * e ≡ S * ðe z Þ has been produced in the article [6] and later studied in [7] (iv) The family S * c ≔ S * ðΛðzÞÞ with ΛðzÞ = 1 + ð4/3Þ z + ð2/3Þz 2 was contributed by Sharma and his coauthors [8] which contains function f ∈ A such that z f ′ðzÞ/f ðzÞ is located in the region bounded by the cardioid given by…”

On Analytic Multivalent Function Associated with Cosine Hyperbolic Function

“…Further, by choosing particular ψ, several subclasses of starlike functions have been studied. See the details in [2,4,5,11,12,14,19]. Recently, Ali et al [1] have acquired adequate conditions on α such that 1 + ϰℊ ′ (ϰ) ℊ (ϰ) ⁄ < √1 + ϰ ⟹ ℊ(ϰ) < √1 + ϰ , for = 0, 1, 2.…”