1. Introduction and Deﬁnitions
Let denote a class of all analytic functions of the form
which are analytic in the open unit disk and normalized by
Let denote the class of functions of the Form (1.1) then if , that is the real part of its first derivative map the unit disk onto the right half plane, then the class of functions in are called functions of bounded turning.
By Nashiro Warschowski, see  , it is proved that the functions in are univalent and also close to convex in U. In  , it was also shown that the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning. For more works on bounded turning see   .
If and are analytic in U, then their Hadamard product defined by the power series is given by:
Note that the convolution so defined is also analytic in U.
For ƒ of the Form (1.1) several interesting derivatives operators in their different forms have been studied, here we consider (1.1) using the modified Caputo’s derivative operator , see   , stated as follow:
For , (1.3)
where is a real number and . Notice that (1.3) can also be express as:
and its partial sum given as:
We determine conditions under which the partial sums of the operator given in (1.4) are of bounded turning. We shall use the following lemmas in the sequel to establish our result.
Lemma 1. 
For , we have
Lemma 2. 
Let P(z) be analytic in U, such that P(0) = 1, and in U. For
function Q analytic in U the convolution function takes values in the convex hull of the image U under Q.
We shall implore lemmas 1 and 2 to show conditions under which the m-th partial sum (2.1) of the modiﬁed Caputoes derivative operator of analytic univalent functions of bounded turning is also of bounded turning.
2. Main Theorem
Let be of the Form (1.1), if and , then , .
Let be of the Form (1.1) and . This implies that
Now for we have
Applying the convolution properties to , where
with recourse for Lemma 1 and we have
Relating Lemma 1 and with , a computation gives
Recall the power series
satisfies and . Therefore by Lemma 2 we have
This proves our results.
 Jahangiri, J.M. and Farahmand, K. (2004) Partial Sums of Functions of Bounded Turning. International Journal of Mathematics and Mathematical Sciences, 2004, Article ID: 451028.
 Darus, M. and Ibrahim, R.W. (2010) Partial Sums of Analytic Functions of Bounded Turning with Applications. Computational & Applied Mathematics, 29, 81-88.
 Salah, J. and Darus, M. (2010) A Subclass of Uniformly Convex Functions Associated with a Fractional Calculus Operator Involving Caputos Fractional Di?erentiation. Acta Universitatis Apulensis. Mathematics-Informatics, 24, 295-306.