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

1. INTRODUCTIONLet [[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 [15], Raina and Srivastava [18] and Aouf [3]);

(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 [10]. The operator [mathematical expression not reproducible] contains as special cases the multiplier transformation [mathematical expression not reproducible] (see Aouf and Hossen [4]), I(m,l) (see Cho et al. [8, 9]) and [I.sup.m] (see Uralegaddi and Somanatha [20] and [21]);

(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 [14]);

(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 [1] and [5]);

(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 [13] and [23]);

(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 [6].

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 [4].

In this paper, known results of Bajpai [7], Goel and Sohi [11], Uralegaddi and Somanatha [20] and Aouf and Hossen [4] 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. [12] 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 [16], 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, [2], [22], [23], 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

[1] 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.

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

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

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

[5] 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.

[6] 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.

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

[8] 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.

[9] 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.

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

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

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

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

[14] 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.

[15] 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.

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

[17] 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.

[18] 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.

[19] 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.

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

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

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

[23] 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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Author: | Ali, Ekram Elsayed; Bulboaca, Teodor |
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Publication: | Tamsui Oxford Journal of Information and Mathematical Sciences |

Date: | Dec 1, 2018 |

Words: | 2820 |

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