Upper bound of fractional differential operator related to univalent functions

In this article, we deﬁned the generalized fractional differential Tremblay operator in the open unit disk that by usage the deﬁnition of the generalized Srivastava– Owa operator. In particular, we established a new operator denoted by H b ; s ; c z based on the normalized generalized fractional differential operator and represented by convolution product. Moreover, we studied the coefﬁcient criteria of univalence, starlikeness and convexity for the last operator mentioned.


Introduction
Let AðmÞ denoted the class of functions wðzÞ of the form: which symbolized by K k ðmÞ K 0 ðmÞ KðmÞ and K k ðmÞ S Ã k ðmÞ. The classes S Ã k ðmÞ and K k ðmÞ have been discussed by many researchers (see [1,2]). For m ¼ 1, the classes K k ð1Þ and S Ã k ð1Þ of order k ð0 k\1Þ were studied before by Robertson [3], and by setting k ¼ 0, they are represented as equivalent form: Theorem 1 (Bieberbach's Conjecture [4,5]) The functions wðzÞ which is defined in (1), is the univalent function in class Sð1Þ, if ja j j j for all j ! 2 and its convex functions in the class Kð1Þ if ja j j 1: Next, the concept of convolution (or Hadamard product) for two analytic and univalent functions wðzÞ given by (1) and hðzÞ ¼ z þ P 1 j¼mþ1 b j z j ; m ¼ f1; 2; 3; . . .g defined by Let us here recall some the well known geometric properties for the convolution (or Hadamard product) due to Ruscheweyh (see [6] (2) The derivative convolution of two functions belong to the class AðmÞ is defined as: (3) Let the functions wðzÞ 2 S Ã ðmÞ and hðzÞ 2 KðmÞ; then ðw Ã hÞðzÞ 2 S Ã ðmÞ. (4) For each functions wðzÞ and hðzÞ 2 KðmÞ; then ðw Ã hÞðzÞ 2 KðmÞ: In [8,9], Srivastava and Owa defined the fractional integral and differential operators in the complex z-plane C as the formula: The fractional integral of order r is defined, for a function f(z) by: z aðgþ1Þþq : Next, we included the Fox-Wright function, which is one of the special functions that generalize hypergeometric functions (see [10]), let denoted this function by p K q and defined as: where q i ; k j are parameters in complex plan C. A i [ 0, B j [ 0 for all j ¼ 1; . . .; q and i ¼ 1; . . .; p, such that 0 1 þ P q j¼1 B j À P p i¼1 A i for fitting values jzj\1: For all z 2 C and j 2 f2; 3; 4; . . .g, the Pochhammer symbol ðzÞ j defined as: where ðzÞ j ¼ CðzþjÞ CðzÞ and the formula CðzÞ is the well known gamma function. In fact, this function have many remarkable properties in complex plan, we here review some of them. For z 2 C, then Moreover, we consider the Bloch space BðUÞ of all functions analytic and univalent functions f in A which is defined as [17]: In the present paper, the generalized Tremblay operator with univalent function Sð1Þ, which is considered as the generalized fractional derivative operator in Definition 5, was defined. After ward, we utilized the normalized generalized Tremblay operator in a class of analytic functions AðmÞ, with subclasses SðmÞ; S Ã k ðmÞ and K k ðmÞ in the open unit disk. Furthermore, we performed some applications to prove the bound coefficient for the last operator.

Results
In this section, we defined the generalized fractional differential of the Tremblay operator in Definition 6 according to definition of the generalized fractional derivative of the Srivastava-Owa operator in complex plane C, for the special case, m ¼ 1 in classes A and S. Examples of power function in complex z-plane and some boundedness properties in Bloch space for the operator mentioned were presented as well.
Definition 6 Let 0 b 1, 0 s 1 and c ! 0. The generalized fractional differential Tremblay operator of two parameters, is defined as where the function f(z) is analytic and univalent in simpleconnected region of the complex z-plane C containing the origin, and the multiplicity of ðz cþ1 À f cþ1 Þ Àbþs is removed by requiring logðz cþ1 À f cþ1 Þ to be non-negative when ðz cþ1 À f cþ1 Þ [ 0: Next, we provided a survey of the interest operator T b;s;c z to satisfy a boundedness property in the open unit disk and gave an example by using Definition 6. Note that proving the boundedness operator on Bloch space requires using expression (1), when m ¼ 1: Example 2 Let f ðzÞ :¼ z j ; z 2 U and j 2 N: If 0\b 1; 0\s 1; c ! 0; and 0 b À s\1; then the generalized of Termblay operator with power function satisfy now go back to Example 1, we see that, if c ¼ 0; then In next theorem, we considered the form of definition of the power series to prove the operator T b;s;c z is bounded with the univalent function S on Bloch space BðUÞ in the open unit disk.

Theorem 2
Let the function f 2 Sð1Þ S belongs to U: Proof By supposing f(z) in class of S, we employ Lemma 1 and Example 2, we obtain where M :¼ r ð1þsÀbÞc ðcþ1Þ bÀs CðsÞ CðbÞ 2 K 1 ðrÞ and 2 K 1 ðrÞ :

Normalized operator
In this section we defined a new operator in Theorem 3, which is normalized for the generalized Tremblay operator T b;s;c z f ðzÞ with an analytic function in the class AðmÞ.
Theorem 3 Let the following conditions to be realized: Then the normalized of generalized Tremblay operator in Definition 6 is denoted by H b;s;c z f ðzÞ and defined as: For all f ðzÞ2 AðmÞ and jzj\1; where Proof From Definition 6, and by considering the function ) which equals to By using the property of gamma function, we have

Criteria for Hadamard product
In this section, the operator in (17) is represented as the convolution product of two univalent functions in class of SðmÞ 2 U, in particular, when m ¼ 1.
Then the proof is completed. h Based on the results, the following observations were obtained. Let f ðzÞ 2 Að1Þ U. Note here the proof of the following Theorems comes immediately from Eq. (20) and Lemma 1.
Theorem 5 Let 0\b 1; 0\s 1 and the condition (16). If the function f(z) given by (1) in the class S Ã ðmÞ and the function g(z) defined by (22) in KðmÞ: Then f ðzÞ Ã gðzÞ 2 S Ã ðmÞ: Theorem 6 Let 0\b 1; 0\s 1 and the condition (16). If the functions f(z) given by (1) Proof By supposing the function f 2 S with equality (17), we have where w j :¼ # b;s;c ðjÞa j and the function # b;s;c ðjÞ defined in (18) satisfied the following condition in class S as follows: j # b;s;c ðjÞ ja j j\1; By using Theorem 1, we give the estimate for the coefficients of an univalent function belong to S in U also, by employ this estimate, we can get another estimate for ' 1 in S as follows, where 'ðjÞ ¼ The series in (23) is transformed into a sum of twice the terms by employing the following relation: Depending on ð1Þ j ¼ j! and ð1Þ jÀ1 ¼ ðj À 1Þ!; the estimate (23) becomes the next form: jþ2 ð1Þ j by considering some properties of the gamma function in (13) and (12), we have and by using the Fox-Wright function, we can transform the estimate ' 1 at z ! 1; We conclude from the above theorem that the operator then the operator maps a convex function f(z) into a univalent function that is H b;s;c z : K ! S: Proof Presume that f ðzÞ 2 K; z 2 U and the operator (17), such that s;c ðjÞa j and the function # b;s;c is defined in inequality (18), satisfied the following condition in class S as follows: We know That the coefficient of a convex function belong to S is ja j j\1. So we can get another estimate for ' 2 as follows, where 'ðjÞ ¼ where a j is Pochhammer symbol defined in (11),with the following relation and ð1Þ j ¼ j!; then the estimate (26) become as the next form jþ1 ð1Þ j employ the properties of gamma function, we have then with the Fox-Wright function, we transform the estimate ' 1 at z ¼ 1; hence Math Sci (2016) 10:167-175 173 H b;s;c for m ¼ f1; 2; 3; . . .g:

Example 3
The function belongs to the class S Ã k ðmÞ; is defined as We prove a bound coefficient in Theorem (10) by using similar methods in the starlike class.

Conclusion
All results of the present work are valid in open unit disk U with respect to the fractional calculus in a complex domain. We defined a normalized fractional differential operator in the concept of the generalized Tremblay operator. Moreover, we assumed sufficient conditions for this operator to become starlike and convex functions. Finally, univalency and convolution properties are discussed.
Author's contributions All the authors jointly worked on deriving the results and approved the final manuscript.