# FAMILIES OF MEROMORPHIC FUNCTIONS INVOLVING A NEW GENERALIZED DIFFERENTIAL OPERATOR.

1. INTRODUCTION

Let [[SIGMA].sub.p,n,] with n [greater than or equal to] 1 - p, denote the class of functions of the form

(1.1) [mathematical expression not reproducible]

which are analytic in the punctured open unit disc U*:= U \ {0}, where U:= {z [member of] C: |z| < 1}.

For a function f [member of] [[SIGMA].sub.p,n] given by (1.1) and g [member of] [[SIGMA].sub.p,n] defined by

[mathematical expression not reproducible]

we define the Hadamard (or convolution) product of f and g by

[mathematical expression not reproducible]

For the complex parameters [[alpha].sub.1],..., [[alpha].sub.q] and [[beta].sub.1],...,[[beta].sub.s], with [mathematical expression not reproducible], let consider the generalized hypergeometric functio[n.sub.q] [F.sub.s]([[alpha].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],..., [[beta].sub.s]; z) defined by (see, for example, [19, p.19])

[q.sub.F.sub.s]([[alpha].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],x x x,[[beta].sub.s]; z) = [[infinity] summation over k = 0] ([[[alpha].sub.1].sub.k]) x x x ([[alpha].sub.q])/[[[beta].sub1].sub.k] x x x ([[beta].sub.s].sub.k]) [z.sup.k]/[k!],z [member of] U,

(q [less than or equal to] s + 1, q,s [member of] [N.sub.0]:= N [union] {0}),

where [([theta]).sub.v] is the Pochhammer symbol, defined, in terms of the Gamma function [GAMMA], by

[mathematical expression not reproducible]

Corresponding to the function [h.sub.pqs]([[alpha].sub.1], x x x,[[alpha].sub.q]; [[beta].sub.1],x x x,[[beta].sub.s]; z) defined by

[h.sub.pqs]([[alpha].sub.1], x x x,[[alpha].sub.q]; [[beta].sub.1],x x x,[[beta].sub.s]; z):= [z.sup.-p] x q[F.sub.s]([[alpha].sub.1], x x x,[[alpha].sub.q]; [[beta].sub.1],x x x,[[beta].sub.s]; z),

we consider a linear operator

[h.sub.pqs]([[alpha].sub.1], x x x,[[alpha].sub.q]; [[beta].sub.1],x x x,[[beta].sub.s]): [[summation].sub.p,n] [right arrow] [[summation].sub.p,n]

which is defined by the following Hadamard product:

[H.sub.p,q,s]([[alpha].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],...,[[beta].sub.s])f (z) = [h.sub.pqs]([[alpha].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],...,[[beta].sub.s]; z) * f (z), z [member of] U*.

Therefore, for a function f of form (1.1), we have

[H.sub.pqs]([[alpha].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],...,[[beta].sub.s])f (z) = [z.sup.-p] + [[infinity] summation over k = n] ([[[alpha].sub.1].sub.k+p]) x x x ([[alpha].sub.q])/[[[beta].sub1].sub.k+p] x x x ([[beta].sub.s].sub.k+p]) [a.sub.k]/[k+p] [z.sup.k],z [member of] U

and, for convenience, we write

[H.sub.pqs]([[alpha].sub.1]:= [H.sub.pqs] [[beta].sub.1],...,[[alpha].sub.q]; [[beta].sub.1],...,[[beta].sub.s]).

Using the Hadamard product, we define the new operator [mathematical expression not reproducible] as follows:

Definition 1.1. For [[alpha].sub.1],...,[[alpha].sub.q] [member of] C and [[beta].sub.1],...,[[beta].sub.s] [member of] C \ [mathematical expression not reproducible], with q [LESS THAN OR EQUAL TO] s + 1 (q, s [member of] [N.sub.0]), let define the operator [mathematical expression not reproducible] by

[mathematical expression not reproducible],

[mathematical expression not reproducible]

[mathematical expression not reproducible]

for m [member of] N, where [alpha], [beta], [lambda] [member of] C and l [member of] N.

Remarks 1.1. (i) We have

[mathematical expression not reproducible]

and for all f [member of] [[summation].sub.p,n] the next formula holds:

(1.2)

[mathematical expression not reproducible].

(ii) For [beta]([lambda] - [alpha]) = 0 or m = 0 in (1.2), we have [mathematical expression not reproducible].

(iii) It easily verified from (1.2) that for all f [member of] [[summation].sub.p,n] we have

(1.3)

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

We emphasize here some special cases of the operator [mathematical expression not reproducible] previously studied by different authors:

(i) [mathematical expression not reproducible] (see Liu and Srivastava , Raina and Srivastava  and Aouf );

(ii) For q = s + 1, [[alpha].sub.i] = 1 (i = 1,...., s + 1) and [[beta].sub.j] = 1 (j = 1,..., s), [alpha] = 0 and [beta] =1, we obtain the operator [mathematical expression not reproducible] introduced and studied by El-Ashwah . The operator [mathematical expression not reproducible] contains as special cases the multiplier transformation [mathematical expression not reproducible] (see Aouf and Hossen ), I(m,l) (see Cho et al. [8, 9]) and [I.sup.m] (see Uralegaddi and Somanatha  and );

(iii) For m = 0, q = 2, s =1, [[alpha].sub.1] = a, [[alpha].sub.2] = 1 and [[beta].sub.1] = c, we have [mathematical expression not reproducible] (a,c > 0) (see Liu and Srivastava );

(iv) For m = 0, q = 2, s = 1, [[alpha].sub.1] = v + p, [[alpha].sub.2] = p and [[beta].sub.1] = p, we have [mathematical expression not reproducible] (see  and );

(v) For m = 0, q = 2 and s = 1, [[alpha].sub.1] = [micro], [[alpha].sub.2] = 1 and [[beta].sub.1] = [micro] + 1, we have [mathematical expression not reproducible] (see  and );

(vi) For m = 0, q = 2, s =1, [[alpha].sub.1] = [lambda] ([lambda] > 0), [[alpha].sub.2] = 1 and [[beta].sub.1] = n + p, we have [mathematical expression not reproducible], where the operator [I.sub.n+p-1,[lambda]] was introduced by Aouf and Xu .

Also, by specializing the parameters m, l, [alpha], [beta], [lambda], p, q, s, [[alpha].sub.i] (i = 1,...., q) and [[beta].sub.j] = 1 (j = 1,..., s) we obtain various new operators:

(i) For q = 2 and s = 1, [[alpha].sub.1] = n + p, [[alpha].sub.2] = 1 and [[beta].sub.1] = 1, we obtain a new operator [mathematical expression not reproducible] where n > - p, p, n [member of] N;

(ii) For q = 2 and s = 1, [[alpha].sub.1] = a, [[alpha].sub.2] = 1 and [[beta].sub.1] = c, we obtain a new operator [mathematical expression not reproducible], where a [member of] R, c [member of] R \ [mathematical expression not reproducible];

(iii) For q = 2 and s = 1, [[alpha].sub.1] = p +1, [[alpha].sub.2] = 1 and [[beta].sub.1] = n + p, we obtain a new operator [mathematical expression not reproducible], where n [member of] Z, n > - p, p, n [member of] N;

(iv) For q = 2 and s = 1, [[alpha].sub.1] = p + [delta], [[alpha].sub.2] = c and [[beta].sub.1] = a, we obtain a new operator [mathematical expression not reproducible], where a,c [member of] [mathematical expression not reproducible], [delta] > -p, p,n [member of] N;

(v) For q = 2 and s = 1, [[alpha].sub.1] = p + [delta], [[alpha].sub.2] = 1 and [[beta].sub.1] = p + [delta] + 1, we obtain a new operator [mathematical expression not reproducible], where [delta] > -p, p, n [member of] N.

We introduce the class [mathematical expression not reproducible] of the functions f [member of] [[summation].sub.p,n] which satisfy the condition

(1.4) [mathematical expression not reproducible]

where

(1.5) [alpha], [lambda] [member of] C with [lambda] [not equal to] [alpha], [beta] [member of] C* l [member of] N, 0 [less than or equal to] [eta] < p, p [member of] N, m [member of] [N.sub.0].

Remark that for the function that appeared in the brackets in the left-hand side of (1.4), the point [z.sub.0] = 0 is a removable singularity, hence this function is regular in the whole unit disk U.

We note that for the special case [lambda] = [beta] = l =1, [alpha] = 0 and q = s + 1, [[alpha].sub.i] = 1 (i = 1,..., s + 1) and [[beta].sub.j] = 1 (j = 1,..., s), the class [mathematical expression not reproducible]([[alpha].sub.i], [alpha], [beta], [lambda], l, [eta]) reduces to the class [B.sub.n]([beta]) studied by Aouf and Hossen .

In this paper, known results of Bajpai , Goel and Sohi , Uralegaddi and Somanatha  and Aouf and Hossen  are extended.

2. BASIC PROPERTIES OF THE CLASS [mathematical expression not reproducible]

We begin by recalling the following well-known result (Jack-Miller-Mocanu's Lemma), which we shall apply in proving our first inclusion theorems.

Lemma 2.1.  Let the nonconstant function w be analytic in U, with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r < 1 at a point [z.sub.0] [member of] U, then

zow'([z.sub.0]) = [rho]w([z.sub.0]),

where [rho] is a real number and [rho] [greater than or equal to] 1.

A generalization of of this lemma was given in , and represents one of the most important investigation tools of the theory of differential subordinations (see also [17, p. 19]).

Unless otherwise mentioned, we assume throughout this paper that all parameters satisfy the conditions (1.5) of the definition formula (1.4).

Theorem 2.1. If we assume that [beta]([lambda] - [alpha]) > 0, then

[mathematical expression not reproducible].

Proof. Considering an arbitrary function [mathematical expression not reproducible], then

(2.1) [mathematical expression not reproducible]

and we have to show that (2.1) implies the inequality (1.4).

Defining the function w regular in U by

(2.2) [mathematical expression not reproducible]

then w(0) = 0, and the above relation may be written as

(2.3) [mathematical expression not reproducible]

Differentiating (2. 3) logarithmically with respect to z and using (1. 3), we obtain

(2.4) [mathematical expression not reproducible]

Now, we will prove that | w(z)| < 1 for z [member of] U If not, then there exists a point [z.sub.0] [member of] U such that max {| w(z)|:|z| [less than or equal to] | [z.sub.0]|} = | w([z.sub.0])| = 1 According to Lemma 2 1, there exists a real number [rho] [greater than or equal to] 1, such that [z.sub.0]w'([z.sub.0]) = [rho]w([z.sub.0]), and taking z = [z.sub.0] in (2 4) we get

[mathematical expression not reproducible]

Since w([z.sub.0]) = [e.sup.i[theta]] for some [theta] [member of] [0, 2[pi]], from the above relation it follows that

[mathematical expression not reproducible]

for all [theta] [member of] [0, 2[pi]], whenever [beta]([lambda] - [alpha]) > 0. This inequality contradicts our assumption given by (2 1), and therefore we have | w(z)| < 1 for all z [member of] U Finally, from (2 2) we conclude that f [member of] [mathematical expression not reproducible]([[alpha].sub.i], [alpha], [beta], [lambda], l, n).

For a number c > 0, let recall the well-known integral operator [F.sub.c>p]: [[summation].sub.p,n] [RIGHT ARROW] [[summation].sub.p,n] defined by

(2.5) [mathematical expression not reproducible]

Remark that the operator [F.sub.c>p] was investigated by many authors, for example, , , , etc. Moreover, for all f G [[summation].sub.c,p] the operator can be written in the convolution product form

[mathematical expression not reproducible]

and we could easily check that it satisfy the following differentiation relationships:

(2.6) [mathematical expression not reproducible]

(2.7) [mathematical expression not reproducible]

Theorem 2.2. If [mathematical expression not reproducible], then [mathematical expression not reproducible], thai is

[mathematical expression not reproducible].

Proof. For an arbitrary f [member of] [[summation].sub.p,n], since the right-hand sides of (2.6) and (2.7) coincide, we get

[mathematical expression not reproducible]

From this last relation, it follows that the assumption [mathematical expression not reproducible] given by (1.4) is equivalent to

(2.8) [mathematical expression not reproducible]

and we have to prove that (2.8) implies the inequality

[mathematical expression not reproducible]

Defining the function w regular in U by

(2.9) [mathematical expression not reproducible]

then w(0) = 0, and the definition relation (2.9) may be written as

(2.10) [mathematical expression not reproducible]

Differentiating (2.10) logarithmically with respect to z and using (2.7), we obtain

[mathematical expression not reproducible]

Using the above relation, the function that appeared in the left-hand side of (2.8) may be written in the form

[mathematical expression not reproducible]

which, by using (2.9) and (2.10), reduces to

(2.11) [mathematical expression not reproducible]

Like in the proof of the previous theorem, we will show that |w(z)| < 1 for z [member of] U. Contrary, there exists a point [z.sub.0] [member of] U such that max{|w(z)|: |z| [less than or equal to] |[z.sub.0]|} = |w([z.sub.0])| = 1, and form Lemma 2.1 there exists a real number [rho] > 1, such that [z.sub.0]w'([z.sub.0]) = [rho]w([z.sub.0]). Thus, using the fact that w([z.sub.0]) = [e.sup.i[theta]] for some [theta] [member of] [0, 2[pi]], and taking z = [z.sub.0] in (2.11) we get

[mathematical expression not reproducible]

for all [theta] [member of] [0, 2[pi]]. Since this inequality contradicts the assumption (2.8), it follows that |w(z)| < 1 for all z [member of] U, and from (2.9) we get our conclusion.

Remarks 2.1. (i) For q = s + 1,[[alpha].sub.i] = 1 (i = 1,..., s + 1), [[beta].sub.j] = 1 (j = 1,..., s), [alpha] = 0, [lambda] = [beta] = l = P = c = [a.sub.k] = 1 and m = [eta] = 0, we note that Theorem 2.2 extends a results of Bajpai [7, Theorem 1];

(ii) For q = s + 1, [[alpha].sub.i] = 1 (i = 1,..., s + 1), [[beta].sub.j] = 1 (j = 1,..., s), [alpha] = 0, [lambda] = [beta] = l = P = [a.sub.k] = 1 and m = [eta] = 0, we note that Theorem 2.2 extends a results of Goel and Sohi [11, Corollary 1].

Theorem 2.3. If we suppose that c = l/[beta]([[lambda]-alpha]) > 0, then [mathematical expression not reproducible] if and only if [mathematical expression not reproducible].

Proof. Differentiating the definition relation (2.5), we get

[mathematical expression not reproducible],

therefore

(2.12) [mathematical expression not reproducible]

Using (1.3), the relation (2.12) becomes

[mathematical expression not reproducible]

and according to the assumption c = l/[beta]([[lambda]-alpha]), it follows that

(2.13) [mathematical expression not reproducible]

From (2.13) we deduce that

[mathematical expression not reproducible]

and our conclusion follows immediately.

Conflict of Interests. The authors declare that there is no conflict of interests regarding the publication of this paper.

REFERENCES

 M. K. Aouf, New certeria for multivalent meromorphic starlike functions of order alpha, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), no. 3, 66-70.

 M. K. Aouf, Certain classes of meromorphic multivalent functions with positive coefficients, Math. Comput. Model., 47(2008), no. 3-4, 328-340.

 M. K. Aouf, Certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric function, Comput. Math. Appl., 55 (2008), 494-509.

 M. K. Aouf and H. M. Hossen, New certeria for meromorphic p -valent starlike functions, Tsukuba J. Math., 17 (1993), no. 2, 481-486.

 M. K. Aouf and H. M. Srivastava, A new criterion for meromorphically p-valent convex functions of order alpha, Math. Sci. Res. Hot-Line, 1 (1997), no. 8, 7-12.

 M. K. Aouf and N. E. Xu, Some inclusion relationships and integral-preserving properties of certain subclasses of p-valent meromorphic functions, Comput. Math. Appl., 61 (2011), no. 3, 642-650.

 S. K. Bajpai, A note on a class of meromorphic univalent functions, Rev. Roum. Math. Pures Appl., 22 (1977), 295-297.

 N. E. Cho, O. S. Kwon, and H. M Srivastava, Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations, J. Math. Anal. Appl., 300 (2004), no. 2, 505-520.

 N. E. Cho, O. S. Known, and H. M. Srivastava, Inclusion relationships for certain subclasses of meromorphic functions associated with a family of multiplier transformations, Integral Transforms Spec. Funct., 16 (2005), no. 8, 647-659.

 R. M. El-Ashwah, A note on certain meromorphic p-valent functions, Appl. Math. Lett., 22 (2009), 1756-1759.

 R. M. Goel and N. S. Sohi, On a class of meromorphic functions, Glas. Mat., 17 (1982), 19-28.

 I. S. Jack, Functions starlike and convex of order [alpha], J. London Math. Soc., (2) 3 (1971), 469-474.

 V. Kumar and S. L. Shukla, Certain integrals for classes of p -valent meromorphic functions, Bull. Aust. Math. Soc., 25 (1982), 85-97.

 J.-L. Liu and H. M. Srivastava, A linear operator and associated with the generalized hypergeometric function, J. Math. Anal. Appl., 259 (2000), 566-581.

 J.-L. Liu and H. M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Model., 39 (2004), no. 1, 21-34.

 S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 289-305.

 S. S. Miller and P. T. Mocanu, Differential Subordination. Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, 2000.

 R. K. Raina and H. M. Srivastava, A new class of meromorphiclly multivalent functions with applications to generalized hypergeometric functions, Math. Comput. Model., 43 (2006), 350-356.

 H. M. Srivastava and P. W. Karlsson, Multiple Gausian Hypergeometric Series, Halsted Press, Ellis Horwood Limited, Chichester, John Wiley and Sons, New York, Chichester, Brisbane and London, 1985.

 B. A. Uralegaddi and C. Somanatha, New criteria for meromorphic starlike univalent functions, Bull. Aust. Math. Soc., 43 (1991), 137-140.

 B. A. Uralegaddi and C. Somanatha, Certain differential operators for meromorphic functions, Houston J. Math., 17 (1991), no. 2, 279-284.

 B. A. Uralegaddi and C. Somanatha, Certain classes of meromorphic multivalent functions, Tamkang J. Math., 23 (1992), 223-231.

 D.-G. Yang, On a class of meromorphic starlike multivalent functions, Bull. Inst. Math. Acad. Sinica, 24 (2) (1996), 151-157.

EKRAM ELSAYED ALI AND TEODOR BULBOACA

Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt

Faculty of Mathematics and Computer Science, Babes,-Bolyai University, 400084 Cluj-Napoca, Romania

E-mail address: ekram 008eg@yahoo.com, E-mail address: bulboaca@math.ubbcluj.ro

Key words and phrases. Meromorphic functions, Hadamard (convolution) product, generalized hypergeometric functions, integral operator.

Received February, 13, 2018, Accepted December, 3, 2018

2010 Mathematics Subject Classification. 30C45, 33C20.
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