Starlike integral operators

THEOREM 1. Let Φ(z) - H and let ψ{z) = H be analytic functions defined in U, with the property φ(z) Φ(z) ΦQ there. Let a, β9 7, and δ be real constants such that (1) a ^ 0, β > 0 , (2) δ^ 0 , (3) a + δ> 0 , and (4) a + δ = β + 7 . // there exists a nonnegative constant J that satisfies

COROLLARY 1. Let Φ{z) = H and φ{z) = H be analytic functions defined in the disc U, with φ(z)-Φ(z) Φ 0 there, and let α, β, 7, δ9 and J be real constants satisfying the conditions a > 0, (a + δ) = (β + 7), and (23) δ + <p(z) Φ(z) in U. Now if feS*, then the function

THEOREM 2. // β > 0, 7 ^ 0 and feS*, then the function F(z) defined by (24) F(z) = Γ£±-ZΓ fWdtΎ β = 3 + • L 2 r Jo J is again an element of S*.

THEOREM 3. If 0 ^ a <* 1, a <; β, and if feS*, then the func- tion (26) F(z) == ί z βA L dt Ύ β = J ( Jo \ t ' is again an element of £*.

Theorem 1 is used. For example, the following example illustrates the idea, used in (A) and (B) above, and it provides an extension of a result due to Merkes and Wright [5]. (C) If 0 ^ a ^ 1, and if feS*, then the function

THEOREM 4. If a > 0, η ^ 0, 7 + rj ^ 0, and if feS*, then the function (29) F{z) = \ a L = z is also starlike in U. For 7 + 37 = 1, a = 1, ^ = 0, 1, 2, , we obtain the sequence [ cz ηi/(Λ+i) 2«-

THEOREM 5. Let a, β, 7, δ, J, λ(J), φ(z), and Φ(z) satisfy the hy- pothesis of Theorem 1, except that (7) is replaced by the condition °L + § + Re ^ 2 M ^ max[0, J - λ(J)] . 2 9>(z) If feK, then FeS*, ^/^erβ F(«) is defined by (9).

Theorem 1. It is worth noting that in such a proof, we use the fact that feS*(l/2), which follows from the hypothesis that feK. Hence the conclusion of Theorem 5 is valid for all feS*(1/2). It is possible to imitate examples (A), (B), and (C) above to show that if feK (or if feS*(1/2)) then the transform of / is smoother than /, for some transforms. For example, (D) if 7 > 0, fe K, then the function F(z) = 1 + Ί\ Z f(t)t r~

THEOREM 6. Let a, β, 7, δ, and p be real constants satisfying the conditions, α Ξ> 0, β > 0, a + δ — β + δ>0 and δ, 7 ^ 0 , Π < / ) < Γ 1 / R Ύ \ Ί

COROLLARY 2. Let a, β, 7, δf and p be real constants such that a > 0, a + δ = β + 7, 0 ^ ρ/2 ^ min[δ, δ - 7]. // fe S*, g e K, then F(z) defined by (30) is a starlike function. Causey and White obtained this last result, but for β — 1, 2, and (β + 7) = 1, 2, . However, our method shows more, at least for β = 1, in which case the transform F(z) is smoother that /(«) — as in examples (A), (B), and (C) above. It is worth noting that Theorem 6 must be used because Corol- lary 2 may not be applicable

THEOREM 7. Let a> β, 7, δ, and p he real constants such that α ^ O , β>0, a + δ = β + Ύ>0 and 0 ^ p ^ 2δ + a , 7 ^ 0 , minΓ 2δ + a, 2δ + a - 2Ί + — minΓ-^ , ^-ΊΊ , 7 > 0 L 2 L7 /SJJ hold. If feK, g e K, then the function F(z), defined in (30) is a starlike function. A proof of this result could be given, one that looks like all the others in this note. We do remark that our result is a little