Abstract
We apply the general theory of differential subordination to obtain certian interesting criteria for -valent starlikeness and strong starlikeness. Some applications of these results are also discussed.
Highlights
Let Ap p ∈ N {1, 2, 3, . . .} be the class of functions f z of the form ∞f z zp ap mzp m m1 which are analytic in the open unit disk Δ : {z : |z| < 1}
F z zp ap mzp m m1 which are analytic in the open unit disk Δ : {z : |z| < 1}
If p z ∈ P satisfies Rp z > 0 z ∈ Δ, we say that p z is a Caratheodory function
Summary
F z zp ap mzp m m1 which are analytic in the open unit disk Δ : {z : |z| < 1}. Let P be the class of functions p z of the form pz 1 pnzn n1 which are analytic in Δ. In this paper by extending the result of Ravichandran and Jayamala 3 , we find sufficient conditions for the subordination p z ≺ q z to hold for given q z and criteria for p-valent starlikeness. Let q z be convex univalent in Δ and satisfy. Let 0 / q z be univalent in Δ and satisfy the following conditions for z ∈ Δ:. By taking α β 0, ξ λ/μ μ > 0, λ > −μ/2, η 1, and q z 1 z / 1 − z in Theorem 2.5 we get the result of Nunokawa et al 2 which was proved by a different method. Letting α β 0, ξ η 1, p z zf z /f z , and q z Theorem 2.5, we get the following. International Journal of Mathematics and Mathematical Sciences Let h u iv, where u and v are real.
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