An Application of S ˘al ˘agean Operator Concerning Starlike Functions

: As an application of the well-known S˘al˘agean differential operator, a new operator is introduced and, using this, a new class of functions S n ( α ) is deﬁned, which has the classes of starlike and convex functions of order α as special cases. Original results related to the newly deﬁned class are obtained using the renowned Jack–Miller–Mocanu lemma. A relevant example is given regarding the applications of a new proven result concerning interesting properties of class S n ( α ) .


Introduction and Preliminaries
Many operators have been used since the beginning of the study of analytic functions. The most interesting of these are the differential and integral operators. Since the beginning of the 20th century, many mathematicians, especially J.W. Alexander [1], S.D. Bernardi [2] and R.J. Libera [3], have worked on integral operators. It has become easier to introduce new classes of univalent functions with the use of operators. In his article, published in 1983, Sȃlȃgean introduced differential and integral operators, which bear his name. Those operators were very inspiring and many mathematicians have obtained new, interesting results using these operators. In particular, researchers have introduced many new operators, examined their properties, and further used the newly defined operators to introduce classes of univalent functions with remarkable properties. At the same time, some mathematicians obtained interesting results in different lines of research by combining differential and integral operators, where Sȃlȃgean differential operator was involved, as is seen, for example, in very recent papers [4][5][6]. The topic of strong differential subordination was also approached recently using Sȃlȃgean differential operator in [7], and new operators were introduced using a fractional integral of Sȃlȃgean and Ruscheweyh operators in [8]. The operators introduced using the Sȃlȃgean differential operator were also recently used to obtain results related to the celebrated Fekete-Szegö inequality [9].
In this work, we introduce a new class as an application of the Sȃlȃgean operator and discuss some interesting problems with this class.
Let A be the class of functions f of the form which are analytic in the open unit disc U = {z ∈ C : |z| < 1} and S be the subclass of A consisting of univalent functions. Also, is the class of starlike functions of order α and is the class of convex functions of order α. Let us start by recalling the well-known definitions for the Sȃlȃgean differential and integral operators.
Definition 1 (Sȃlȃgean [10]). For f ∈ A, the Sȃlȃgean differential operator D n is defined by and Sȃlȃgean integral operator D −n is defined by and k n a k z k (n = 1, 2, 3, · · · ).
With the above operator D j f , we introduce the subclass S n (α).
For a function f ∈ A, we introduce For the above M p (r, f ), we define To discuss our problems, we have to introduce the following lemmas. [12]).
Discussing our problems for Sȃlȃgean operator, we need to introduce the following lemma due to Miller and Mocanu [17,18] (also, by Jack [19]). [17,18]). Let the function w given by

Lemma 7 (Miller and Mocanu
be analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a point z 0 ∈ U, then a real number k ≥ n exists, such that and The original results obtained by the authors and presented in this paper are contained in the next section. A new operator is introduced with Sȃlȃgean differential operator as the inspiration. Using this newly introduced operator, a new class of functions denoted by S n (α) is defined, with known classes as particular cases. Certain properties involving the applications of Sȃlȃgean differential operator related to class S n (α) are discussed in the theorems and corollaries. Examples are also included to prove the applications of the proved results.

Main Results
Now, we derive the following result.
Further, if then there exists δ > 0, such that D n−j f ∈ H Proof. We note that if f ∈ S n (α), then Re where α 0 = α. Since and we see that Re Applying Lemma 1, we say that This implies that that is, that f ∈ S n−j (α j ). Further, applying Lemma 2, we see that if then there exists δ > 0, such that D n−j f ∈ H

Example 1. Let us consider a function f belonging to the class S 3 (α). Then f ∈
Further, f ∈ S 1 (α 2 ), where If we consider the case of α = 1 4 , then we have and Further, if we consider the case of α = 1 8 , then and Remark 2. For some positive integer j, we know that If we consider g(0) = 1 2 and g(1) = 0. From this fact, we know that α j < α j+1 for 0 ≤ α j < 1. This implies that Letting j = n in Theorem 1, we see then there exists δ > 0, such that f ∈ H δ+ 1 2(1−α j ) .

Next we have
for some n ∈ N, then there exists p j , such that D n−j+1 f ∈ H p j , where and j ≤ n + 1.
Proof. If we define p by then p is analytic in U with p(0) = 1. Since we see that Re Applying Lemma 3, we have that Using Lemma 4, we know that that is, that (D n−1 f (z)) ∈ H p 1 . By Lemma 6, we have that Noting that we obtain that Repeating the above, we have that Finally, we get D n−j+1 f ∈ H p j (0 < p j < 1).
Next, we derive for some real α ( 1 2 ≤ α < 1), then D n f ∈ S 0 (α), that is, D n f is starlike of order α in U. Further, if then, there exists δ > 0 such that D n−j f ∈ H and j ≤ n.
Proof. Define a function w by It follows from the above that Therefore, we have that Suppose that there exists a point z 0 ∈ U, such that Then, Lemma 7 say that w(z 0 ) = e iθ and z 0 w (z 0 ) = kw(z 0 ) (k ≥ 1). This implies that This contradicts our condition of the theorem. Thus we say that |w(z)| < 1 for all z ∈ U. From the definition (57) for w, we obtain that This means that D n f ∈ S 0 (α). Letting α = α 0 and using Lemma 1, we obtain D n−j f ∈ S 0 (α j ), where α j is given by (56). Applying Lemma 2, we know that if then, there exists δ > 0 such that D n−j f ∈ H Making j = n in Theorem 3, we have for some real α ( 1 2 ≤ α < 1), then D n f ∈ S 0 (α). If then, there exists δ > 0, such that f ∈ H δ+ 1 2(1−αn ) .

Conclusions
Inspired by the classic and well-known Sȃlȃgean differential operator, a new operator is introduced in Definition 2. By applying this operator, a new class of functions is defined, denoted by S n (α). It is shown that classes of starlike and convex functions of the order α are obtained for specific values of n. Some interesting problems concerning the class S n (α) are discussed in the theorems and corollaries. One example is given as an application for special cases of n for the class S n (α). The new operator defined in this paper can be used to introduce other certain subclasses of analytic functions. Quantum calculus can be also associated for future studies, as can be seen in paper [20] regarding the Sȃlȃgean differential operator and involving symmetric Sȃlȃgean differential operator in paper [21]. Symmetry properties can be investigated for this operator, taking the symmetric Sȃlȃgean derivative investigated in [22] as inspiration.