Radius properties for analytic and p-valently starlike functions
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Let be the class of functions f(z) which are analytic in the open unit disk and satisfy . Also, let denotes the subclass of consisting of f(z) which are p-valently starlike of order α(0 ≦ α < p). A new subclass of is introduced by
for some real λ > 0. The object of the present paper is to consider some radius properties for such that .
2010 Mathematics Subject Classification: Primary 30C45.
KeywordsAnalytic radius problem Cauchy-Schwarz inequality p-valently starlike of order α.
for some real number λ > 0.
This shows that for λ ≧ 1.
Which shows that for λ ≧ 5.
which shows that for λ ≧ 11.
for some real α (0 ≦ α < p).
A function is said to be p-valently starlike of order α in (cf. Robertson ).
2 Coefficient inequalities
For , we consider the sufficient condition for f(z) to be in the class .
Therefore, we say that .
Next, we discuss the necessary condition for the class .
Remark 1 If we take p = 1 in Lemmas 1 and 2, then we have that
3 Radius problems
Our main result for the radius problem is contained in
This completes the proof of the theorem.
for λ > 0. Therefore, |δ0(λ)| given by (3.2) is increasing for λ > 0.
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