Sufficient Conditions for Strong Starlikeness
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Sufficient Conditions for Strong Starlikeness Kanika Sharma1 · Nak Eun Cho2 · V. Ravichandran3 Dedicated to Prof. Dato’ Indera Rosihan M. Ali Received: 7 July 2020 / Accepted: 18 August 2020 © Iranian Mathematical Society 2020
Abstract Let p be an analytic function defined on the open unit disc D with p(0) = 1 and 0 < α ≤ 1. The conditions on complex valued functions C, D, and E are obtained for p to be subordinate to ((1 + z)/(1 − z))α when C(z)z 2 p (z) + D(z)zp (z) + E(z) p(z) = 0. Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to ((1 + z)/(1 − z))α are obtained as applications. The conditions on α and β are derived for p to be subordinate to ((1+z)/(1−z))α when 1+βzp (z)/ p n (z) with n = 1, 2 is subordinate to 1 + 4z/3 + 2z 2 /3 =: ϕC A R (z). Similar problems were investigated for Re p(z) > 0 when the function p(z) + βzp (z)/ p n (z) with n = 0, 2 is subordinate to ϕC A R (z). The condition on β is determined for p to be subordinate to ((1 + z)/(1 − z))α when p(z) + βzp (z)/ p n (z) with n = 0, 1, 2 is subordinate to ((1 + z)/(1 − z))α . Keywords Starlike functions · Differential subordination · Confluent hypergeometric function · Bessel function Mathematics Subject Classification 30C80 · 30C45
Communicated by Ali Abkar.
B
Nak Eun Cho [email protected] Kanika Sharma [email protected] V. Ravichandran [email protected]; [email protected]
1
Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi, Delhi 110 021, India
2
Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea
3
Department of Mathematics, National Institute of Technology, Tiruchirappalli 620 015, India
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Bulletin of the Iranian Mathematical Society
1 Introduction For a natural number n, let H[a, n] be the class of all analytic functions p defined on the open unit disc D of the form p(z) = a + pn z n + pn+1 z n+1 + · · · . An analytic function p ∈ H[1, 1] is a function with a positive real part if Re p(z) > 0 and the class of all such functions is denoted by P. Let An = {zh : h ∈ H[1, n]} and let A := A1 . Let S denote the subclass of A consisting of univalent (one-to-one) functions. For an analytic function ϕ with ϕ(0) = 1, let: S ∗ (ϕ) :=
f ∈A:
z f (z) ≺ ϕ(z) . f (z)
This class unifies various classes of starlike functions when Re ϕ > 0. Shanmugam [23] studied the convolution properties of this class when ϕ is convex, while Ma and Minda [13] investigated the growth, distortion, and coefficient estimates under less restrictive assumption that ϕ is starlike and ϕ(D) is symmetric with respect to the real axis. For 0 ≤ α < 1, the class S ∗ ((1 + (1 − 2α)z)/(1 − z)) =: S ∗ (α) is the class of starlike functions of order α, introduced by Robertson [21]. The class S ∗ := S ∗ (0) is the familiar class of starlike functions. For 0 < α ≤ 1, S ∗ [α] := S ∗ (((1+z)/(1−z))α ) is the class of strongly starlike functions of order α. For 0 ≤ β < 1,
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