Abstract
Let Qα(α ≥0) denote the class of normalized analytic alpha-quasi-convex functions
f, defined in the unit disc, D = {z : |z| < 1}, by the condition
Re
h
(1 −α) f ′(z)
g′(z) + α (zf ′(z))′
g′(z)
i
> 0,
Where f(z) = z +P∞
n=2 anzn and where g(z) = z +P∞
n=2 bnzn is a convex univalent function
in D. Sharp upper bounds are obtained for |a3 −µa2
2|, when µ ≥0.
Results & Lemmas (2)
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Lemma 1
Lemma 1 ([5], p.166). Let h ∈p, i.e., let h be analytic in D and satisfy Re h(z) > 0 for z ∈D, with h(z) = 1 + c1z + c2z2 + · · · Then c2…
Lemma 1 ([5], p.166). Let h ∈p, i.e., let h be analytic in D and satisfy Re h(z) > 0 for z ∈D, with h(z) = 1 + c1z + c2z2 + · · · Then c2 −c2 1 2 ≤2 −|c1|2 2 .
Lemma 2
Lemma 2 ([2]). Let g ∈C, with g(z) = z + b2z2 + b3z3 + · · ·, then for µ real, |b3 −µb2 2| ≤max(1/3, |1 −µ|).
Lemma 2 ([2]). Let g ∈C, with g(z) = z + b2z2 + b3z3 + · · ·, then for µ real, |b3 −µb2 2| ≤max(1/3, |1 −µ|).
Function classes studied:
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