The radius of starlikeness for a class of regular functions defined by an integral

Theorem 1 provides a lower bound for the radius of /3-starlikeness of ^Γ(α, 7, c) and Theo- rem 3 gives the radius of /3-starlikeness of <^(α, 7, c). We begin by stating a slight generalization of the result obtained by Lewandowski et al mentioned above. Since our result follows directly from the techniques used in [8], the proof will be omitted.

LEMMA 1. // F(z) and f(z) satisfy (2), f(z) is in S*(a) and c ^ 0, then F{z) is in S*(a). This lemma now enables us to determine a lower bound for the radius of /3-starlikeness of <^Γ(α, 7, c).

THEOREM 1. If F(z) is in J^ia, 7, c), then F(z) is β-starlike for \z\ < σ = σ(a, β, 7, c), where σ is the least positive root of the equation ( 5 ) 1 - β - r[2(l - a) + 2c(l - 7)] - r\2a - 1 - β + 2c(l - 7)] = 0 .

LEMMA 2. // ω(z) is analytic and satisfies \<o(z)\ ^ \z\ in E and

LEMMA 3. // p(z) and a)(z) satisfy the conditions of Lemma 2, then for any K ;> B we have on \z\ — r Re \κp(z) 1 ^ p(«)ί (P^r) /or Ro ^ R, (P2(r) for Ro ^ Λj where PM = P^r, K, B, D) = P.2{r) = P,{r, K, B, D)

THEOREM 2. min min Re , :„(«,,•,.> l«|=r ( f{z) ( 6 ) (7 ) 2Z? - - K) 1 — D H 1 K-1 + 2D

THEOREM 3. Let r* = r*(a, y, c, β) be the radius of β-starlike- ness of J^2(a, y, c). Let D = 2d - 1, δ = (a + cy)/(l + c), c ^ 0, 0 ^ α < 1, αm£ 0 ^ 7 ^ 1. i^or eαc/^ ^ e d c in [0, 00), Ze£ r{D) be the