We define the function which is a partial sum of by
(2)
Theorem 1
The function
satisfies
(3)
for
Furthermore, for
Proof
Noting that
it follows that for , we obtain
Moreover, we also observe that
Now assume that
for
This completes the proof. □
Remark 2 For example, the values , and imply the radius of starlikeness of is , and for the same values, the radius of starlikeness of the ordinary partial sums is (see [2]).
Next, we derive the radius of convexity.
Theorem 3
The function
satisfies
(4)
for
Furthermore, for
Proof
A computation gives
Therefore, for , we obtain
Moreover, we impose
Now, consider that
for
This completes the proof. □
Remark 4 In view of Theorem 3, for example, the values , and pose the radius of convexity of is and for the same values, the radius of convexity of the ordinary partial sums is (see [2]).
Next, we assume special ordinary partial sums depending so that their coefficients satisfy the relation .
Theorem 5
Assume the partial sum
Then the function .
Proof We consider α such that
This implies that
that is,
By letting , we define the function as follows:
Logarithmic derivative of yields
where
Now, for all and , the function has unique real negative zeros in the interval . This leads to the fact that has unique positive real zeros for all distributed in the interval . Therefore, we will calculate α in . It is easy to check that is decreasing for in the interval . Moreover, we have
We conclude that for ,
thus . This completes the proof. □
By letting in Theorem 5, we have the following result.
Corollary 6
The Cesáro partial sums
of the function are starlike of order .
Theorem 7 Assume the partial sum as in Theorem 5. Then the function .
Proof We consider α such that
This implies that
therefore, a computation gives
thus
By putting , we define the function as follows:
Logarithmic derivative of yields
where
Now, for all and , the function has unique real negative zeros in the interval . This leads to the fact that has unique positive real zeros for all in the interval . So, we calculate α in the interval . A computation yields is decreasing for in the interval . Thus, we have
which implies . This completes the proof. □
Theorem 8
Assume the Cesáro partial sum
of the function
Then the function for all .
Proof We consider α such that
This implies that
therefore, a computation gives
thus
By putting , we define the function as follows:
Logarithmic derivative of yields
The function has a unique real negative zero in the interval for all which is around . This leads to the fact that has a unique positive real zero in the interval around . A computation yields is decreasing in the interval and assuming its maximums at and . Thus, we have
which implies . This completes the proof. □
Note that some other results related to partial sums can be found in [11–15].