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

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...