Radius Problems For Functions Associated with a Nephroid Domai

Lemma 2.1. Let 1/3 < a < 5/3, and let ra be given by ra =    a −1/3 if 1/3 < a ≤1, 5/3 −a if 1 ≤a < 5/3. Then {w ∈C : |w −a| < ra} ⊆ΩNe, where ΩNe := ϕNe(D), the region bounded by the nephroid (1.4).

Lemma 3.1 ([18, Lemma 2.1, p. 267]). If p ∈P[A, B], then for |z| = r < 1, p(z) −1 −ABr2 1 −B2r2 ≤(A −B)r 1 −B2r2 .

Theorem 3.1. (i) Let 0 ≤B < A ≤1. Then the S∗ Ne-radius for S∗[A, B] is given by RS∗ Ne (S∗[A, B]) = min  1, 2 3A −B  . In particular, if 1 −B ≤3(1 −A), then S∗[A, B] ⊂S∗ Ne. (ii) Let −1 ≤B < A ≤1 with B ≤0. Then the S∗ Ne-radius for S∗[A, B] is given by

Theorem 3.3]. Specializing the constants A and B in Theorem 3.1, the following important sharp radii results are obtained.

Corollary 3.1. The sharp S∗ Ne-radius for S∗(α) = S∗[1 −2α, −1] is 2/(3(1 −2α) + 5). The estimate is sharp for the function kα(z) = z(1 −z)2α−2, 0 ≤α < 1.

Corollary 3.2. The sharp S∗ Ne-radius for the class of starlike functions S∗is 1/4, and the sharpness holds for the well-known Koebe function k(z) := z/(1 −z)2.

Corollary 3.3. The sharp S∗ Ne-radius for the convex class C is 2/5.

Theorem 3.2. For the function classes BS∗(α), S∗ L(α), and S∗ α,e, the following radius results hold: (i) For α ∈[0, 1), RS∗ Ne (BS∗(α)) = ρB(α) := 4/(3 + √9 + 16α). (ii) For α ∈[0, 1/3], RS∗ Ne (S∗ L(α)) = ρL(α) := 4(2 −3α)/9(1 −α)2. In particular, RS∗ Ne (S∗ L) = ρL := 8/9. (iii) For α ∈[0, (3e −5)/(3e −3)], RS∗ Ne

Theorem 3.3. For the function classes S∗ RL, S∗ C and S∗ R, we have: (i) RS∗ Ne (S∗ RL) = ρRL := 56/(122 −41 √ 2) ≈0.874764, (ii) RS∗ Ne (S∗ C) = ρc := √ 2 −1 ≈0.414214, (iii) RS∗

Theorem 3.4. The S∗ Ne-radii for the classes S∗$ and S∗ S are given by (i) RS∗ Ne  S∗$  = ( √ 17 −1)/6 ≈0.520518, (ii) RS∗ Ne (S∗ S) = sinh−1(2/3) = log 

Lemma 4.1 ([18],[20]). Let p ∈Pn(α). Then, for |z| = r, the following results hold.

Theorem 4.1. For the function classes G1, G2, G3 and G4, we have the following radii results. (i) RS∗ Ne,n (G1) = ρ1 :=  3n + √ 9n2 + 1 −1/n. (ii) RS∗ Ne,n (G2) = ρ2 := n√ 4 ×  9n +

Theorem 4.2. The sharp S∗ Ne,n-radius for CS∗ n(α) is given by RS∗ Ne,n (CS∗ n(α)) = ρcs :=   2 3(1 + n −α) + q 9(1 + n −α)2 + 4(4 −3α)   1/n

Lemma 4.1(b), we have

Theorem 4.3. The sharp S∗ Ne,n-radius for Wn is RS∗ Ne,n (Wn) = ρW :=  2/(3n + √ 9n2 + 4) 1/n .

Theorem 4.4. The S∗ Ne,n-radius for the class Mn(β) is given by RS∗ Ne,n (Mn(β)) = (3β −2)−1/n. The result is sharp for the function f(z) = z(1 −zn)2(β−1)/n. References [1] R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218 (2012), no. 11, 6557–6565. [2] R. M. Ali, N. K. Jain and V. Ravichandran, On the radius constants for classes of analytic functions, Bull. Malays. Math. Sci. Soc. (2) 36