On starlikeness of certain integral transforms

Theorem 1. If δ = (2 ln 2 −1)(3 −2 ln 2)/[3 −(2 ln 2 −1)(3 −2 ln 2)] = 0.262 . . . and f ∈R(−δ) then the function F defined by (1) is starlike in U. P r o o f. From (1) we deduce (6) zF ′′(z) + F ′(z) = f ′(z), z ∈U . Let P(z) = F ′(z) and Q(z) = F(z)/z. Since Re f ′(z) > −δ in U, by using (4) and (6) we find that Re F ′(z) > −δ +(1+δ)(2 ln 2−1) for z ∈U. Again by using (4) this in turn implies Re[Q(z)] > 2δ > 0, z ∈U. Now if we set p(z) = zF ′(z)/F(z) then p is analytic in U, p(0) = 1 and f ′(z)

Lemma 1. If g ∈R(β) then G defined by (2) belongs to R(β +(1−β)(3− 4 ln 2)) (β < 1). P r o o f. From (2) we deduce zG′(z) + G(z) = 2g(z) , (7) zG′′(z) + 2G′(z) = 2g′(z) . (8) Since g ∈R(δ), by using (5), we obtain G′(z) ≺β + (1 −β)L(z), z ∈U , and so Re G′(z) > β + (1 −β)(3 −4 ln 2), z ∈U. Here L(z) is as defined earlier. This proves Lemma 1.

Lemma 2. Let M = 2(2 ln 2 −1)(1 −ln 2), θ = 0.911621907, N = tan θ, a = 4(1+M)2−4 3N 2(2 ln 2−1)4−4, b = −4(1−2M)(1+M)−8 3(2 ln 2−1)4N 2, c = (1−2M)2 −2 3(2 ln 2−1)2N and ϱ = (−b−(b2 −4ac)1/2)/(2a). Suppose that Q is a complex function with Q(0) = 1 satisfying (9) Q(U) ⊂E1 ∩E2 ∩E3 where E1 = {w ∈C : Re w > 1 −2M(1 + ϱ)} , E2 = {w ∈C : |arg (w −(1 −2(2 ln 2 −1)(ϱ + 1)))| < θ} , E3 = {w ∈C : |Im w| < 2(2 ln 2 −1)(ϱ + 1)π} . If p is analytic in U with p(0) = 1 and if Re Q(z)[zp′(z) + p2(z) + p(z)]

Theorem 2. Let ϱ be as defined in Lemma 2, i.e., ϱ ≈0.09032572 . . . and g ∈R(−ϱ). Then the Libera transform G defined by (2) is in S∗. P r o o f. Since g ∈R(−ϱ), by using Lemma 1 we obtain G ∈R(β) with (13) β = −ϱ + (1 + ϱ)(3 −4 ln 2) = 1 −2(2 ln 2 −1)(ϱ + 1) . Now using (4) and the fact that G ∈R(β) we get (14) (G(z)/z) ≺β + (1 −β)l(z), z ∈U , where l(z) = −1 −(2/z) log(1 −z). By (13), a simple calculation yields β + (1 −β)(2 ln 2 −1) = 1 −2M(1 + ϱ). This, from (14) and the observation made earl

Theorem 3. If h ∈A satisfies Re{h′(z)h(z)/z} > −ϱ in U then the function H defined by H(z) = R z 0 (h(t)/t) dt is starlike in U. R e m a r k 2. In [6], the author showed that for f ∈A and 1/6 ≤β < 1, Re[h′(z)h(z)/z] > β((3β −1)/2) implies Re(f(z)/z) > β in U.

Theorem 4. If f ∈A satisfies Re[f ′(z)] > (−2ϱ(2 + α) + 1 −2α)/5, z ∈U, for α ≥1/2, then the function F defined by F(z) = αz1−1/α zR o f(t)t1/α−2 dt is in S∗. Corollary. If f ∈A satisfies Re f ′(z) > −(6ϱ + 1)/5 ≈0.3083908 . . . for z in U, then the function F defined by (1) is starlike in U. The above corollary improves Theorem 1. R e m a r k 4. For g defined by g(z) = z(2 + z)/2(1 −z) (and hence g satisfies Re[zg′(z)/g(z)] > −1/2 in U) it is well known that the correspond- ing Libera transform G is