Starlike And Convex Functions Associated with A Nephroid domain having Cusps On The Real Axis

Theorem 1.1. The function ϕNe(z) = 1 + z −z3/3 maps D onto the region bounded by the nephroid  (u −1)2 + v2 −4 9 3 −4v2 3 = 0, (1.2) which is symmetric about the real axis and lies completely inside the right-half plane u > 0 (see Figure 1).

Lemma 2.1. For 0 < r < 1, the function ϕNe(z) satisfies min |z|=r Re  ϕNe(z)  =    1 −r + 1 3r3, if r ≤ 1 √

Lemma 2.2. Let 1/3 < a < 5/3. Let ra and Ra be given by ra =    a −1 3, 1 3 < a ≤1 5 3 −a, 1 ≤a < 5 3. and Ra = 1

Theorem 2.1. A function f belongs to the class S∗ Ne if, and only if, there exists an analytic function q(z), satisfying q(z) ≺ϕNe(z) = 1 + z −z3/3 such that f(z) = z exp Z z 0 q(ζ) −1 ζ dζ ! , z ∈D. (2.1)

Corollary 2.1. A function f belongs to the class CNe if, and only if, there exists an analytic functions q(z), satisfying q(z) ≺ϕNe(z) such that f(z) = Z z 0

Theorem 2.2. Let f ∈S∗ Ne and let |z| = r < 1. Then (i) Subordination results: zf′(z) f(z) ≺ zf′ Ne(z) fNe(z) and f(z) z ≺fNe(z) z . (ii) Growth Theorem: −fNe(−r) ≤|f(z)| ≤fNe(r). Equality holds for some non-zero z if, and only if, f is a rotation of fNe(z). (iii) Covering Theorem: Either f is a rotation of fNe or f(D) contains the disk ∆∗=

Theorem 2.3. Let f ∈CNe and let |z| = r < 1. Then (i) Subordination results: zf′(z) f(z) ≺zb f′ Ne(z) b fNe(z) and f(z) z ≺b fNe(z) z . (ii) Distortion Theorem: bf ′

Theorem 3.1. The function class S∗ Ne satisfies the following inclusion properties: (a) S∗ Ne ⊂S∗(α) whenever 0 ≤α ≤1 −2 √ 2 3 ≈0.057191. (b) S∗ Ne ⊂SS∗(β) whenever β0 ≤β ≤1, where β0 ≈0.929121. (c) S∗ qc ⊂S∗ Ne whenever 0 < c ≤8 9. (d) k −ST ⊂S∗

Theorem 3.2. The class CNe satisfies the following properties: (a) CNe ⊂C(α), whenever 0 ≤α ≤1 −2 √ 2 3 . (b) CNe ⊂SC(β), whenever β0 ≤β ≤1, where β0 ≈0.929121. (c) C(qc) ⊂CNe, whenever 0 < c ≤8 9, where C(qc) is the convex class corresponding to the class S∗ qc. (d) k −UCV ⊂CNe, whenever k ≥5 2. (e) Cα,e ⊂CNe, whenever α ≥1 −2/3 e−1, where Cα,e is the convex class corresponding to S∗

Lemma 3.1 ([25, Lemma 2.1]). If p ∈P[A, B], then for |z| = r < 1 p(z) −1 −ABr2 1 −B2r2 ≤(A −B)r 1 −B2r2 .

Theorem 3.3. Let −1 < B < A ≤1. Then S∗[A, B] ⊂S∗ Ne, if the parameters A and B satisfy any one of the following conditions: (1 −B2) < 3(1 −AB) ≤3(1 −B2) and (1 −B) ≤3(1 −A). (3a) 3(1 −B2) ≤3(1 −AB) < 5(1 −B2) and 3(1 + A) ≤5(1 + B). (3b)

Corollary 3.1. Let −1 < B < A ≤1. If either of the conditions (3a)–(3b) mentioned in

Theorem 3.3 hold true, then the Janowski convex class C[A, B] is contained in CNe. 4. Coefficient Estimates

Theorem 4.1. Let f(z) = z + P∞ n=2 anzn be a member of the function class S∗ Ne. Then ∞ X n=2  (3n)2 −72 |an|2 ≤40.

Corollary 4.1. Let f(z) = z + P∞ n=2 anzn ∈S∗ Ne. Then |an| ≤ s 40 (3n)2 −72 for n = 3, 4, .... Using the fact that f ∈CNe ⇐⇒ zf ′(z) ∈S∗ Ne, the following result is an immediate consequence of Theorem 4.1.

Theorem 4.2. Let f(z) = z + P∞ n=2 anzn ∈CNe. Then ∞ X n=2 n2  (3n)2 −72 |an|2 ≤40.

Corollary 4.2. Let f(z) = z + P∞ n=2 anzn ∈CNe. Then |an| ≤1 n s 40 (3n)2 −72 for n = 3, 4, ....

Theorem 4.3. (i) Let Fn(z) = z + anzn (n = 2, 3, ...). Then Fn ∈S∗ Ne if, and only if, |an| ≤ 2 3n −1, n = 2, 3, .... (ii) The function KA(z) = z/(1 −Az)2 belongs to S∗ Ne if, and only if, |A| ≤1/4. (iii) Let |b| < 1. Then Lb(z) = zebz ∈S∗ Ne if, and only if, |b| ≤2/3.

Corollary 4.3. (i) z + anzn ∈CNe if, and only if, |an| ≤ 2 n(3n −1), n = 2, 3, .... (ii) z/(1 −Az) ∈CNe if, and only if, |A| ≤1/4. (iii) (ebz −1)/b ∈CNe if, and only if, |b| ≤2/3. For z ∈D, define the class of functions P as P := ( p(z) = 1 + ∞ X n=1

Lemma 4.1. Let p(z) = 1 + P∞ n=1 pnzn ∈P. Then [18] |pn| ≤2 for n ≥1, (4.5) and for any complex number ξ, we have |p2 −ξp2 1| ≤2 max {1, |2ξ −1|}. (4.6) The results are sharp for the functions p(z) = 1 + z 1 −z, p(z) = 1 + z2 1 −z2 .

Theorem 4.4. (i) If f(z) = z + a2z2 + a3z3 + · · · ∈S∗ Ne, then |a2| ≤1, |a3| ≤1/2, and for any complex number µ, |a3 −µa2 2| ≤1 2 max {1, |2µ −1|} . In particular, |a3 −a2 2| ≤1/2. Equality holds for the function fNe(z). (ii) If f(z) = z + a2z2 + a3z3 + · · · ∈CNe, then |a2| ≤1/2 and |a3| ≤1/6. Equality holds for the function bfNe(z). Further, for any complex number µ, |a3 −µa2 2| ≤1 6 max 

Lemma 5.1 ([21, Theorem 3.4h, p. 132]). Let q be univalent in D, and let ψ and ν be analytic in a domain D containing q(D) with ψ(w) ̸= 0 when w ∈q(D). Set Q(z) := zq′(z)ψ  q(z)  and h(z) := ν  q(z)  + Q(z). Suppose that either (i) h is convex, or (ii) Q is starlike.

Theorem 5.1. Let p ∈H such that p(0) = 1, and let 1 + βzp′(z) ≺ϕNe(z). Then the following subordinations hold: (a) p(z) ≺ϕ$(z) = √ 1 + z2 + z whenever β ≥4 9  2 + √ 2  ≈1.51743. (b) p(z) ≺ϕC(z) = 1 + 4 3z + 2 3z2 whenever β ≥4$

Corollary 5.1. If f ∈A satisfies 1 + βzf ′(z) f(z)

Theorem 5.2. Let p ∈H satisfies p(0) = 1, and let 1 + βzp′(z) p(z) ≺ϕNe(z). Then the following subordinations hold: (a) p(z) ≺ϕS(z) whenever β ≥ 8/9 log(1+sin 1) ≈1.45585. (b) p(z) ≺ϕ0(z) whenever β ≥− 8/9 log(2 √ 2−2) ≈4.72245. (c) p(z) ≺ϕNe(z) whenever β ≥ 8/9 log(5/3).

Corollary 5.2. If f ∈A satisfies 1 + β

Theorem 5.3. Let p ∈H such that p(0) = 1, and let 1 + βzp′(z) p2(z) ≺ϕNe(z). Then the following subordinations hold: (a) p(z) ≺ϕ$(z) whenever β ≥4 9  2 + √ 2  ≈1.51743. (b) p(z) ≺ϕC(z) whenever β ≥4 3. (c) p(z) ≺ϕlim(z) whenever β ≥$

Corollary 5.3. If f ∈A satisfies 1 + β zf ′(z) f(z) !−1 1 −zf ′(z) f(z) + zf ′′(z) f ′(z) ! ≺ϕNe(z), then (a) f ∈S∗$(z) whenever β ≥4 9  2 +$