J. Aust. Math. Soc. 99 (2015), 315–333

Theorem 2.1. Let f ∈S∗ m(A, B) for some A ∈C, −1 ≤B ≤0 with A , B. Then, for 0 < r ≤1, L1(r, f) ≤  F(δ, δ; 1; B2r2m) for B , 0, 0F1  1; |A|2r2m m2  for B = 0, (2.1) where δ = 1/m(A/B −1). The inequality (2.1) is sharp.

Corollary 3.1d.1, page 76]) z f(z) = p(z) ≺exp Z z 0 φ(t) t dt  = q(m) A,B(z), where q(m) A,B(z) = ((1 + Bzm)(1/m)(A/B−1) = F(1, δ; 1; Bzm)

Lemma 2.4. Let f ∈S∗ m(A, B) for some A ∈C, −1 ≤B ≤0 with A , B. If z f(z) = 1 + ∞ X k=1 bkmzkm for z ∈D, (2.6) then ∞ X k=1 ((km)2 −|B −A −kmB|2)|bkm|2 ≤|A −B|2.

Lemma 2.5. Let f ∈S∗ m(A, 0) for some A ∈C\{0}. If z f(z) = 1 + ∞ X k=1 bkmzkm for z ∈D and q(m) A,0(z) = e−(A/m)zm = 1 + ∞ X k=1

Lemma 2.6. Let f ∈S∗ m(A, B) for some A ∈C, −1 ≤B < 0 with A , B. If z f(z) = 1 + ∞ X k=1 bkmzkm for z ∈D and q(m) A,B(z) = (1 + Bzm)(1/m)(1−A/B) = 1 + ∞ X k=1

Theorem 2.7. Let f ∈S∗ m(A, B) for some A ∈C, −1 ≤B ≤0 with A , B. Then, for 0 < r ≤1, max f∈S∗m(A,B) ∆  r, z f(z)  = E(m) A,B(r), where E(m) A,B(r) =

Corollary 2.9. Let f ∈S∗ m(β) for some 0 ≤β < 1. Then, for 0 < r ≤1, max f∈S∗m(β) ∆  r, z f(z)  = 4π m (1 −β)2r2mF  2 m(β −1) + 1, 2 m(β −1) + 1; 2; r2m .

Corollary 2.10. Let f ∈S∗ m(γ) := S∗ m(1 −2γ, −1) for some γ ∈C\{0}. Then, for 0 < r ≤1, max f∈S∗m(1−2γ,−1) ∆  r, z f(z)  = 4π m |γ|2r2mF 

Corollary 2.11 [21, Theorem 1.3]. Let f ∈S∗((1 −2β)α, −α) for some 0 < α ≤1 and 0 ≤β < 1. Then, for 0 < r ≤1, max f∈S∗((1−2β)α,−α) ∆  r, z f(z)  = 4πα2(β −1)2r2F(2β −1, 2β −1; 2; α2r2). The maximum is attained for the function k(1−2β)α,−α(z) defined by (1.4). If we choose β = 0 in Corollary 2.11, then we obtain the result of Sahoo and Sharma [21, Theorem 3.1]. This solves Yamashita’s conjecture for functions in the class S(α) which was introduced by Padmanabhan [16]. More generally, if we choos

Corollary 2.12 [17, Theorems 2.1 and 2.3]. Let f ∈S∗ 1(A, B, 1) = S∗(A, B) for some −1 ≤B ≤0, A , B and A ∈C. Then, for 0 < r ≤1, max f∈S∗(A,B) ∆  r, z f(z)  = EA,B(r), https://doi.org/10.1017/S1446788715000154 Published online by Cambridge University Press

Corollary 2.7]. This solves the maximal area problem for functions in the class S∗((b2 −a2 + a)/b, (1 −a)/b) which was introduced by Silverman [22]. Acknowledgements The authors thank Professor S. Ponnusamy for useful discussions and thank the referee for useful comments and suggestions. References [1] V. V. Anh, ‘k-fold symmetric starlike univalent functions’, Bull. Aust. Math. Soc. 32(3) (1985), 419–436. [2] J. G. Clunie, ‘On meromorphic Schlicht functions’, J. Lond. Math. Soc. (2) 34 (1959),