Let An be the class of
all analytic functions f of the form
f(z)=z +
∑k=n +
1∞akzk,
z∈Δ,
where n∈N is fixed. For
λ>0 and α<1, define
Un(λ)={f∈An:
|(z/f(z))n +
1f'(z) - 1|<λ,
z∈Δ}
and
Sα∗={f∈S∗(α): |z
f'(z)/f(z) - 1|<1 - α,
z∈Δ}.
In this paper, we find suitable conditions on λ and α so
that Un(λ) is
included in Sα and S∗(α). Here Sα
and S∗(α) denote
the usual classes of strongly starlike and starlike of order α,
respectively. We determine necessary conditions so that
f∈Un(λ)
implies that
|z
f'(z)/f(z) -
1/(2β)|<1/(2β), z∈Δ,
or
|1 + z
f''(z)/f'(z) -
1/(2β)|<1/(2β), |z|<r,
where r=r(λ,n) will be specified.
For c + 1 - n>0, define
[I(f)](z)=F(z)=z[(c
+ 1 - n)/zc + 1 -
n.∫0z(t/f(t))ntc
- n d t]1/n.
We also find conditions on λ, α and c so
that I(Un(λ))⊂Sα∗.
2000
Mathematics Subject Classification: 30C45, 30C55.
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