# Starlike and Convex Functions Associated With Nephroid Curve (Research Paper Sample)

In this paper, the starlike and convex functions associated with nephroid curve are introduced.

the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete–Szegö problems are discussed.

We note that all the regions considered earlier were having either no cusp or a single one (cardioid), but in our case, the region bounded by the nephroid curve consists of two cusps.

1 Introduction

:=∈: | |Let C be the complex plane and D{zCz < 1} be the open unit disk. Let

A∈ A≤SC∈ H=−=SH := HA(D) be the collection of all analytic functions defined on D, and let consist of functions f satisfying f (0) f r(0) 1 0. Further, let be the family of functions f that are univalent in D. For 0 α < 1, let ∗(α) and (α) be the subclasses of which consist of functions that are, respectively, starlike and convex of order α. Analytically, these classes are represented as

f (z)S∗(α) := . f ∈ A : Re . zf r(z) Σ > αΣ and

rf (z)C(α) := . f ∈ A : Re .1 + zf rr(z) Σ > αΣ .

SC⊂ AS = SC = C∈ C∈ S≤SS⊂ AThe classes∗∗(0) and(0) are the well-known classes of starlike and convex functions. These two classes are related by the familiar Alexander’s theorem as: f(α) if and only if zf r∗(α). For 0 <β1, the classes∗(β)and (β), consisting of strongly starlike and strongly convex functions of order β,

are defined as

.. .

. f (z) .2.

SC(β) :=

f : .arg

1 + zf rr(z) Σ. < βπ Σ .

SS∗(β) := . f : .arg zf r(z) . < βπ Σ and.f r(z).2

SS⊂ SSS= SSSObserve that ∗(1) ∗, and for 0 < β < 1, ∗(β) consists only of bounded starlike functions, and hence in this case the inclusion ∗(β) ∗ is proper.

:→=∈⊂≺==≺=⊂∈ H=||∈ H≺For f , g , the function f is said to be subordinate to g, written as f g, if there exists a function w satisfying w(0) 0 and w(z) < 1 such that f (z) g(w(z)). Indeed, f g implies that f (0) g(0) and f (D) g(D). Moreover, if the function g(z) is univalent, then f g if, and only if, f (0) g(0) and f (D) g(D). Here, the function w(z) is the well-known Schwarz function. An analytic function f D C satisfying f (0) 1 and Re ( f (z)) > 0 for every z D is called a Carathéodory function.

Ma and Minda [19] used the concept of subordination to develop an interesting method of constructing subclasses of starlike and convex functions. For this purpose, we consider the analytic function ϕ : D → C satisfying the following:

1 ϕ(z) is univalent with Re(ϕ) > 0,

2 =ϕ(D) is starlike with respect to ϕ(0)1,

3 ϕ(D) is symmetric about the real axis, and

4 ϕr(0)> 0.

Throughout this manuscript, wherever the analytic function ϕ is given, it implies that the function ϕ retains the above properties. Using this ϕ, the authors in [19] defined

the function classes S∗(ϕ) and C(ϕ) as

S∗(ϕ) := . f ∈ A : zf r(z) ≺ ϕ(z)Σ andC(ϕ) := . f ∈ A : 1 + zf rr(z) ≺ ϕ(z)Σ .

f (z)

f r(z)

(1.1)

SSSSC=+−≤S []C[]SC=++∈ [−]+−SC=+ −−≤=SCSCIt is clear that for every ϕ satisfying conditions (i)–(iv), the classes ∗(ϕ) and (ϕ) are the subclasses of∗ and , respectively. Specialization of the function ϕ(z) in (1.1) leads to a number of well-known function classes. For instance, taking ϕ(z) (1 z)/(1 z) yields ∗ and , and taking ϕ(z)(1 (1 2α)z)/(1 z) (0α< 1) yields ∗(α) and (α). If we set ϕ(z)(1Az)/(1Bz), where A, B1, 1 and B < A, we obtain the Janowski classes ∗ A, B andA, B (see [12]). The classes∗(β) and(β) are obtained for ϕ(z)((1z)/(1z))β (0 <β1). The parabolic starlike class P introduced by Rønning [27] and the uniformly convex class UCV introduced by Goodman [10] are obtained for the function

π 2ϕ(z) = 1 + 2 .

1√z 2

Σ+log 1 − √z

, z ∈ D.

≤∞− ST−For 0 k < , the classes k(k-uniformly starlike functions) and k UCV (k- uniformly convex functions) introduced by Kanas and Wisniowska [13] are obtained from (1.1) on taking the function ϕ(z) as

1

ϕ(z) = pk (z) := 1 − k2 cosh

2 cos 1 k

.−log

π

1 √z

.ΣΣ+1 − √z

k2

− 1 − k2 .(1.2)

SL.Σ.Σwith the right-half of the lemniscate of Bernoulli u + v− 2 u − v= 0.Sokół and Stankiewicz [32] introduced and discussed the starlike class ∗ associated

22 222

In Ma–Minda’s form, SL∗ := S∗(√1 + z). Moreover, a function f ∈ SL∗ is called a

Sokół and Stankiewicz starlike fu√nction. Sokół [33] introduced another important class

.ΣD+−cSq∗c := S∗(qc), where qc(z) =1 + cz with c ∈ (0, 1]. For c ∈ (0, 1), the function

.Σ−=−q (z) mapsonto the interior of right loop of the Cassinian ovals u2v2 2

2 u2v2c21. The following Ma–Minda-type classes have been introduced in the recent past:

* The class SR∗ L := S∗(ϕRL) with

ϕRL

(z) = √2 − (√2 − 1), 1 − z,

1 +√2( 2 − 1)zwas considered by Mendiratta et al. [20]. The function ϕRL(z) maps D onto the region enclosed by the left-half of the shifted lemniscate of Bernoulli

.(u − √2)2 + v2Σ2 − 2 .(u − √2)2 − v2Σ = 0.

* maps D onto the crescent-shaped regionThe function class Sg∗

:= S∗(z + √1 + z2) was introduced by Raina and Sokół

* w ∈ C : |w2 − 1| < 2|w|, Re w> 0Σ.The class Se∗ := S∗(ez) was introduced and discussed by Mendiratta et al. [21].

* Sharma et al. [29] introduced and investigated the class SC∗ := S∗(1 + 4z/3 + 2z2/3) associated with the cardioid (9u2 + 9v2 − 18u + 5)2 − 16(9u2 + 9v2 − 6u + 1) = 0, a heart-shaped curve.

* The function class SR∗ := S∗(ϕ0), where ϕ0(z) is the rational function

0ϕ (z) := 1 + z . k + z Σ = 1 + 1 z + 2 z2 + ··· , (k = 1 + √2),(1.3)

[24] and then further discussed in [8,25,3.0]. The function ϕg(z) = z + √1 + z2kk − z

kk2

was discussed by Kumar a√nd Ravichandran [16].

* The class Sl∗i m := S∗(1 + 2z + z2/2) associated with the limacon (4u2 + 4v2 −

−++−−=8u5)28(4u24v212u3)0 was considered by Yunus et al. [35].

* Recently, Kargar et al. [14] discussed the following starlike class associated with the Booth lemniscate:

1 −2αzBS(α) := S∗ .1 + z Σ , 0 ≤ α< 1.

* S := S+Cho et al. [7] introduced the Ma–Minda-type function class S∗∗(1sin z)

associated with the sine function.

* For 0 ≤ α< 1, Khatter et al. [15] introduced and discussed in detail the classes

Sα∗,e := S∗ .α + (1 − α)ez Σ and SL∗ (α) := S∗ .α + (1 − α)√1 + zΣ ,

associated with the exponential function and the lemniscate of Bernoulli. Clearly, for α = 0, these classes reduce to the function classes Se∗ and SL∗ , respectively.

* Very recently, Goel and Kumar [9] introduced the starlike class SS∗G := S∗(ϕSG)

associated with the sigmoid function ϕSG(z) = 2/(1 + e−z).

Motivated by the aforementioned works, in this paper, we introduce the classes SN∗ e := S∗(ϕNe) and CNe := C(ϕNe), where the analytic function ϕNe : D → C is defined as ϕNe(z) := 1 + z − z3/3. First we show that SN∗ e and CNe are well defined, i.e., ϕNe(z) satisfies conditions (i)–(iv). For z1, z2 ∈ D, ϕNe(z1) = ϕNe(z2) gives

(z1 − z2)(z2 + z1z2 + z2 − 3) = 0. This holds true only if z1 = z2, and hence, proves

123

∈=−the univalency of ϕNe(z) in D. Also, for each zD, the function L(z)zz /3 satisfies

Re . zLr(z) Σ = Re . 1 − z2

Σ = 1 − 2 Re .z2Σ

L(z)

1 − z2/3

31 − z2/3

2

≥ 1 −

|z|2

> 1 − 1 = 0

.

3 1 − |z|2/3

= +=This shows that the function L(z) is starlike in D, and consequently, the function ϕNe(z) 1 L(z) is starlike with respect to ϕNe(0) 1. We note that the starlikeness of L(z) in D is also clear in light of the Brannan’s criteria for starlikeness of a univalent

polynomial [5, Theorem 2.3]. The condition ϕNr e(0) > 0 is easy to verify. For the remaining, we prove the following result.

=+ −Theorem 1.1 The function ϕNe(z) 1 z z3/3 maps D onto the region bounded by the nephroid

(u − 1)+ v− 9.22

4 Σ3

−3 = 0,(1.4)4v2

which is symmetric about the real axis and lies completely inside the right-half plane u > 0 (see Fig. 1).

∈ −]Proof It is enough to prove that for t( π, π , the...

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