Third-order differential subordination and superordination involving a fractional operator

Abstract The third-order differential subordination and the corresponding differential superordination problems for a new linear operator convoluted the fractional integral operator with the Carlson-Shaffer operator, are investigated in this study. The new operator satisfies the required first-order differential recurrence (identity) relation. This property employs the subordination and superordination methodology. Some classes of admissible functions are determined, and these significant classes are exploited to obtain fractional differential subordination and superordination results. The new third-order differential sandwich-type outcomes are investigated in subsequent research.

Significant and interesting problems in the geometric function theory are studied using third-order differential subordination and superordination for functions, which are analytic in the unit disk. In 1935, Goluzin [15] considered the simple first-order differential subordination zp 0 .z/ h.z/ and showed that if h is convex, then p.z/ q.z/ D R z 0 h.t/t 1 dt; and this q is the best dominant. In 1970, Suffridge [16, p. 777] improved the Goluzin's result. In 1947, Robinson [17, p. 22] considered the differential subordination p.z/ C zp 0 .z/ h.z/ and showed that if h and q.z/ D z 1 R z 0 h.z/dt are univalent, then q is the best dominant, at least for jzj < 1=5. In 1975, Hallenbeck and Rusheweyh [18] considered the differential subordination p.z/ C zp 0 .z/ h.z/ . 6 D 0; Re 0/ and proved that if h is convex, then p.z/ q.z/ D z R z 0 h.t /t 1 dt; and this is the best dominant. The theory of differential subordination in C is the complex analogue of differential inequality in R: This theory of differential subordination was initiated by the works of Miller and Mocanu in 1981 [19], which was developed in other studies in 1987 [20] and 1989 [21]. Many significant works on differential subordination were pioneered by Miller and Mocanu, and their monograph (2000) [22] compiled their considerable efforts in introducing and developing the same. In 2003, Miller and Mocanu [23] investigated the dual problem of differential superordination, whereas Bulboaca (2005) [24] investigated both subordination and superordination. The theory of first and second order differential subordination and superordination has been used by numerous authors to solve problems in this field (see [25][26][27][28][29][30]). By contrast, few articles mentioned third-order inequalities and subordination. The first authors investigated the third order, and Ponnusamy et al. [31] published in 1992. In 2011, Antonino and Miller [32] extended the theory of second order differential subordination in the open unit disk U introduced by Miller and Mocanu [22] to the third order case. They determined the properties of p functions that satisfy the following third-order differential subordination: f .p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z/ W z 2 Ug : In 2013, Jeyaraman et al. [33] also applied the third-order subordination result on the Schwarzian derivative. In 2014, Tang et al. [34] introduced the concept of the third-order differential superordination, which is a generalization of the second-order differential superordination. They determined the properties of functions that satisfy the following third-order differential superordination: f .p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z/ W z 2 Ug: They also obtained the differential subordination and the corresponding differential superordination implications for meromorphically multivalent functions, which are defined by convolution operators involving the Liu-Srivastava operator by determining certain classes of admissible functions. In 2014, Tang et al. [35] investigated some thirdorder differential subordination results for analytic functions involving the generalized Bessel functions. In 2014, Tang et al. [36] studied the differential superordination based on analytic functions involving the generalized Bessel functions. In 2014, Farzana et al. [37] discussed some third-order differential subordination results for analytic functions which are associated with the fractional derivative operator.
The present study utilized the methods of the third-order differential subordination and superordination results of Antonino and Miller [26] and Tang et al. [34], respectively. Certain suitable classes of admissible functions are considered in this study, and some applications of the third-order differential subordination and superordination of analytic functions associated with the new operator are investigated. Several interesting examples are also discussed.

Preliminaries
Let H.U/ be the class of all holomorphic functions f .z/ which are defined in the unit disk U. For a 2 C and n 2 N, let H OEa; n D ff 2 H.U/ W f .z/ D a C a n z n C a nC1 z nC1 C :::g and also let H 0 D H OE0; 1 and H 1 D H OE1; 1 : Let f and F be members in H.U/, the function f is said to be subordinate to F , or F is superordinate to f , if there is an analytic function g.z/ in U with g.0/ D 0 and jg.z/j < 1 for all z 2 U, such that f .z/ D F .g.z//. In this case, we write f F; or f .z/ F .z/: Furthermore, if the function F is univalent in U; then [22]: LetA denote the well-known class of all normalized holomorphic functions f .z/ of the form f .z/ D z C 1 X nD2 a n z n ; .z 2 U/: The integrals of the second type G˛.z/ .˛2 C/ defined in (1), can be written as follows: na n t n 1 # 2 C ::: 1 ! a n z n C :::; where A n is the combination of˛and a n : Corresponding to the Carlson-Shaffer operator L.˛; c/f .z/ defined by [38] L.˛; c/.f .z// D z C 1 X nD1 .˛/ n .c/ n a n z nC1 ; .z 2 U/ and the integrals of the second type G˛f .z/ defined in (1), we consider a linear operator T˛.˛2 C/ by satisfying the following first-order differential recurrence (identity) relation: for some c˛2 C: The development by which we attain at fractional operators is somewhat similar to what was done for numbers. First, we had positive integers, and then tailed the zero, fractions, irrational, negative, and complex numbers. Nevertheless, the utility of fractional operators and derivatives is wide-ranging. Especially, we will constrain ourselves to the areas of geometric function theory and univalent function theory. For convenience, we will also assume that the numbers and functions treated here are generalizations to complex numbers. The applications of the fractional operators have appeared in various fields such as control theory [39], image processing [40][41][42] and diffusion concept [43].
In this section, we offer each of the essential definitions and fundamental theorems in theory of the third-order differential subordination and superordination which will deal with to derive our major results. We first recall the basic concepts in theory of the third-order differential subordination due to Antonino and Miller [32]. . Let W C 4 U ! C and the function h.z/ be univalent in U. If the function p.z/ is analytic in U and satisfies the following third-order differential subordination .p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z/ h.z/; (4) then p.z/ is called a solution of the differential subordination.
A univalent function q.z/ is called a dominant of the solutions of the differential subordination, or, more simply, a dominant if p.z/ q.z/ for all p.z/ satisfying (5). A dominant e q.z/ that satisfies e q.z/ q.z/ for all dominants q.z/ of (5) is said to be the best dominant. The subordination methodology is applied to an appropriate class of admissible functions. The following class of admissible functions was given by Antonino and Miller [32]. . Let be a set in C and q 2 Q and n 2 Nnf1g. The class of admissible functions ‰ n OE ; q consists of those functions W C 4 U ! C achieving the following admissibility condition: .r; s; t; uI z/ … whenever r D q. /; s D k q 0 . /; < where z 2 U; 2 @UnE.q/; and k n.
The next theorem is the foundation result in the theory of third-order differential subordination.
A univalent subordinant e q.z/ that satisfies the condition q.z/ e q.z/ for all subordinants q.z/ of (6) is said to be the best subordinant, (see [34]). Tang et al. [34] considered the following class of admissible functions related to differential superordination.
Definition 2.6 ( [34]). Let be a set in C; q 2 H OEa; n and q 0 .z/ ¤ 0. The class of admissible functions ‰ 0 n OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: where z 2 U; 2 @U; and m n 2.

Subordination of the integral operator T˛f .z/
In this section, the following class of admissible functions is defined, which is required to prove the main third-order differential subordination theorem for the operator T˛f .z/ defined by (3).
Let be a set in C, c˛; c˛C 1 ; c˛C 2 2 Cnf0g and q 2 Q 0 T H 0 . The class of admissible functionŝ T OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: where z 2 U; 2 @UnE.q/ and k 2.
If the function f 2 A and q 2 Q 0 satisfy the following conditions: and then T˛f .z/ q.z/ .z 2 U/ : Proof. Define the analytic function p.z/ in U by From equations (4) and (10), we have Further computations show that and Define the transformation from C 4 to C by v.r; s; t; u/ D r; w.r; s; t; u/ D s C .c˛ 1/r cx .r; s; t; u/ D t C .c˛C c˛C 1 1/s C .c˛ 1/.c˛C 1 1/r c˛c˛C 1 ; (13) and y.r; Let .r; s; t; uI z/ D .v; w; x; yI z/ D r; The proof will make use of Theorem 2.4. Using equations (10) to (13), and from (16), we have Hence, (9) becomes p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á 2 : The next result is an extension of Theorem 3.2 to the case where the behavior of q.z/ on @U is not known. then T˛f .z/ q.z/ .z 2 U/ : Proof. From Theorem 3.2, yields T˛f .z/ q .z/ .z 2 U/ : The result asserted by corollary 3.3 is now deduced from the following subordination property: q .z/ q.z/ .z 2 U/.
If ¤ C is a simply connected domain, then D h.U/, for some conformal mapping h.z/ of U onto . In this case, the classˆT OEh.U/; q is written asˆT OEh; q. The following result follows immediately as a consequence of Theorem 3.2.
If the function f 2 A and q 2 Q 0 satisfy the following conditions (8) and then The next result is an immediate consequence of Corollary 3.3.
Corollary 3.5. Let C and let the function q be univalent in U with q.0/ D 0. Let 2ˆT OEh; q for some If the function f 2 A and q satisfy the following conditions: The following result yields the best dominant of the differential subordination (18).
Theorem 3.6. Let the function h be univalent in U and let W C 4 U ! C and be given by (16). Suppose that the differential equation q.z/; zq 0 .z/; z 2 q 00 .z/; z 3 q 000 .z/I z has a solution q.z/ with q.0/ D 0, which satisfies condition (8).
If the function f 2 A satisfies condition (18) and .T˛f .z/; T˛C 1 f .z/; T˛C 2 f .z/; T˛C 3 f .z/I z/ is analytic in U, then T˛f .z/ q.z/ and q.z/ is the best dominant.
Proof. From Theorem 3.2, we have q is a dominant of (18). Since q satisfies (19), it is also a solution of (18) and therefore q will be dominated by all dominants. Hence q is the best dominant.
In view of Definition 3.1, and in the special case q.z/ D M z; M > 0, the class of admissible functionsˆT OE ; q; denoted byˆT OE ; M ; is expressed as follows.
Definition 3.7. Let be a set in C; c˛; c˛C 1 ; c˛C 2 2 C n f0g and M > 0. The class of admissible functionŝ whenever z 2 U; < Le iÂ .k 1/kM; and < Ne iÂ 0 for all Â 2 R and k 2. Proof. By taking .v; w; x; yI z/ D w in Corollary 3.9, we have to find the condition so that 2ˆT OEM , that is, the admissibility condition (20)  Definition 3.12. Let be a set in C, q 2 Q 1 T H 1 and c˛; c˛C 1 ; c˛C 2 2 C n f0g. The class of admissible functionsˆT ;1 OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: where z 2 U; 2 @UnE.q/ and k 2.
Theorem 3.13. Let 2ˆT ;1 OE ; q. If the function f 2 A and q 2 Q 1 satisfy the following conditions: and Â T˛f .z/ z ; then T˛f .z/ z q.z/ .z 2 U/ : Proof. Define the analytic function p.z/ in U by p.z/ D T˛f .z/ z : By using equation (4) and (24), we get Further computations show that and Define the transformation from C 4 to C by v.r; s; t; u/ D r; w.r; s; t; u/ D s C c˛r cx .r; s; t; u/ D t C .c˛C c˛C 1 C 1/s C c˛c˛C 1 r c˛c˛C 1 ; and y.r; Let .r; s; t; uI z/ D .v; w; x; yI z/ D r; s C c˛r c˛; The proof will make use of Theorem 2.4. Using equations (24) to (27), and from (30), we obtain p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á D Â T˛f .z/ z ; Hence, (23) becomes p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á 2 : Thus, the admissibility condition for 2ˆT ;1 OE ; q in Definition 3.12 is equivalent to the admissibility condition for 2 ‰ 2 OE ; q as given in Definition 2.3 with n D 2. Therefore, by using (22)  In view of Definition 3.12, and in the special case q.z/ D M z; M > 0, the class of admissible functionsˆT ;1 OE ; q; denoted byˆT ;1 OE ; M ; is expressed as follows. .k C c˛/M e iÂ c˛; .k C c˛/M e iÂ c˛; whenever z 2 U; Â 2 R and k 2: Definition 3.20. Let be a set in C; c˛; c˛C 1 ; c˛C 2 ; c˛C 3 2 C n f0g and q 2 Q 1 T H 1 . The class of admissible functionsˆT ;2 OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: and < Â " where z 2 U; 2 @UnE.q/ and k 2.
Theorem 3.21. Let 2ˆT ;2 OE ; q. If the function f 2 A and q 2 Q 1 satisfy the following conditions: and Â then Proof. Define the analytic function p.z/ in U by By using equation (4) and (36), we get Further computations show that and Define the transformation from Let .r; s; t; uI z/ D .v; w; x; yI z/ The proof will make use of Theorem 2.4. Using equations (36) to (39), and from (42), we obtain p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á D Â Hence, (35) becomes p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á 2 : Note that t s C 1 D c˛C 1 .w 1/ c˛.2v 1/ C If ¤ C is a simply connected domain, then D h.U/, for some conformal mapping h.z/ of U onto . In this case, the classˆT ;2 OEh.U/; q is written asˆT ;1 OEh; q. The following result follows immediately as a consequence of Theorem 3.21.
Theorem 3.22. Let 2ˆT ;2 OEh; q. If the function f 2 A and q 2 Q 1 satisfy the following conditions (34) and Â then In view of Definition 3.20, and in the special case q.z/ D 1 C M z; M > 0, the class of admissible functionŝ T;2 OE ; q; denoted byˆT ;2 OE ; M ; is expressed as follows. (44) whenever z 2 U; < Le iÂ .k 1/kM; and < Ne iÂ 0 for all Â 2 R and k 2.

Superordination of the integral operator T˛f .z/
In this section, the third-order differential subordination theorem for the operator T˛f .z/ defined by (3) is investigated. For this purpose, the class of admissible functions is given in the following definition.
Definition 4.1. Let be a set in C; c˛; c˛C 1 ; c˛C 2 2 C n f0g and q 2 H 0 with q 0 .z/ ¤ 0. The class of admissible functionsˆ0 T OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: .v; w; x; yI / 2 whenever v D q.z/; w D zq 0 .z/ C m.c˛ 1/q.z/ mc˛; implies that q.z/ T˛f .z/ .z 2 U/ : Proof. Let the function p.z/ be defined by (10) and by (16). Since 2ˆ0 T OE ; q; (16) and (47) yield f p.z//; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á W z 2 Ug: From equations (14) and (15), we see that the admissible condition for 2ˆ0 T OE ; q in Definition 4.1 is equivalent to the admissible condition for as given in Definition 2.6 with n D 2: Hence 2 ‰ 0 2 OE ; q; and by using (46) and Theorem 2.7, we have q.z/ p.z/ D T˛f .z/: If ¤ C is a simply connected domain, then D h.U/, for some conformal mapping h.z/ of U onto . In this case, the classˆ0 T OEh.U/; q is written asˆ0 T OEh; q. The following result follows immediately as a consequence of Theorem 4.2. implies that q.z/ T˛f .z/ .z 2 U/ : Theorem 4.2 and 4.3 can only be used to obtain subordinations of the third-order differential superordination of the forms (47) or (48). The following theorem proves the existence of the best subordinant of (48) for a suitable chosen .
Proof. In view of Theorem 4.1 and Theorem 4.3 we deduce that q is a subordinant of (48). Since q satisfies (49), it is also a solution of (48) and therefore q will be subordinated by all subordinants. Hence q is the best subordinant. is univalent in U, and the condition (8)  Definition 4.6. Let be a set in C, c˛; c˛C 1 ; c˛C 2 2 C n f0g and q 2 H 1 with q 0 .z/ ¤ 0. The class of admissible functionsˆ0 T;1 OE ; q consists of those functions W C 4 U ! C that satisfy the following admissibility condition: .v; w; x; yI / 2 whenever v D q.z/; w D zq 0 .z/ C mc˛q.z/ mc˛; Proof. Let the function p.z/ be defined by (36) and by (42). Since 2ˆ0 T;2 OE ; q; (43) and (53) yield f p.z//; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z Á W z 2 Ug: From equations (40) and (41), we see that the admissible condition for 2ˆ0 T;2 OE ; q in Definition 4.10 is equivalent to the admissible condition for as given in Definition 2.6 with n D 2: Hence 2 ‰ 0 2 OE ; q; and by using (52) and Theorem 2.7, we have q.z/ p.z/ D T˛f .z/: If ¤ C is a simply connected domain, then D h.U/, for some conformal mapping h.z/ of U onto . In this case, the classˆ0 T;2 OEh.U/; q is written asˆ0 T OEh; q. The following result follows immediately as a consequence of Theorem 4.11.
Theorem 4.12. Let 2ˆ0 T;2 OEh; q and the function h be analytic in U. If the function f 2 A; T˛f .z/ 2 Q 1 and q 2 H 1 with q 0 .z/ ¤ 0 satisfy the following conditions (52) and

Conclusion
In term of the fractional calculus in a complex domain, we defined a new fractional integral. The above operator is a generalization of several integral operators such as the Carlson-Shaffer operator. This fractional operator may be used to obtain new classes of analytic functions in the open unit disk. Moreover, we introduced a new application of the third-order differential subordination and superordination to obtain a sandwich theorem involving the new fractional integral operator. Fractional inequalities are suggested in this work by utilizing the fractional integral operator of different order. These inequalities have been shown the upper and lower cases of this operator in the unit disk.