• Korean Journal of Mathematics
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• v.31 no.3
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• pp.243-258
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• 2023

A normalized analytic function f defined on the unit disk 𝔻 is starlike of reciprocal order α, 0 ≤ α < 1, if Re(f(z)/(zf'(z))) > α for all z ∈ 𝔻. Such functions are starlike and therefore univalent in 𝔻. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order α. Furthermore, a few conditions are derived for a function f to belong to a subclass of reciprocal starlike functions, satisfying |f(z)/(zf'(z)) - 1| < 1 - α.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.120-135
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• 2024

A normalized analytic function f, defined on the unit disk 𝔻, is starlike of reciprocal order α > 1 if the real part of f(z)/(zf'(z)) is less than α for all z ∈ 𝔻. By utilizing the theory of differential subordination, we establish several sufficient conditions for analytic functions defined on 𝔻 to be starlike of reciprocal order. Additionally, we investigate the conditions under which the function f(z)/(zf'(z)) is subordinate to the function 1 + (α - 1)z. This subordination, in turn, is sufficient for the function f to be starlike of reciprocal order α > 1.