Abstract
Let
f
k
(
z
)
=
z
+
∑
n
=
2
k
a
n
z
n
{f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n}
be the sequence of partial sums of the analytic function
f
(
z
)
=
z
+
∑
n
=
2
∞
a
n
z
n
f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}
. In this paper, we determine sharp lower bounds for
Re
{
f
(
z
)
/
f
k
(
z
)
}
{\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\}
,
Re
{
f
k
(
z
)
/
f
(
z
)
}
{\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\}
,
Re
{
f
′
(
z
)
/
f
k
′
(
z
)
}
{\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\}
and
Re
{
f
k
′
(
z
)
/
f
′
(
z
)
}
{\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\}
, where
f
(
z
)
f\left(z)
belongs to the subclass
J
p
,
q
m
(
μ
,
α
,
β
)
{{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta )
of analytic functions, defined by Sălăgean
(
p
,
q
)
\left(p,q)
-differential operator. In addition, the inclusion relations involving
N
δ
(
e
)
{N}_{\delta }\left(e)
of this generalized function class are considered.
Elementary inclusion relations for generalized numerical ranges