SOME INCLUSION PROPERTIES OF CERTAIN CLASS OF

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of analytic functions defined by using the Dziok-Srivastava operator. Some of ... g(z), if and only if there exists a function ω, analytic in U such that ω(0) = 0, |ω(z)| ..... St. Ruscheweyh, Convolutions in Geometric Function Theory, Sem. Math. ... ing fractional calculus, generalized hypergeometric functions, Hadamard products,.
TAIWANESE JOURNAL OF MATHEMATICS Vol. 13, No. 6B, pp. 2001-2009, December 2009 This paper is available online at http://www.tjm.nsysu.edu.tw/

SOME INCLUSION PROPERTIES OF CERTAIN CLASS OF ANALYTIC FUNCTIONS Jacek Dziok and Janusz Soko´ l/

Abstract. We use a property of the Bernardi operator in the theory of the Briot– Bouquet differential subordinations to prove several theorems for some classes of analytic functions defined by using the Dziok-Srivastava operator. Some of these results we obtain applying the convolution property due to Rusheweyh. We take advantage of the Miller–Mocanu differential subordinations.

1. INTRODUCTION Let A denote the class of functions f of the form: f (z) = z +

(1)

∞ 

an z n

n=2

which are analytic in U = U (1), where U(r) = {z : z ∈ C and |z| < r}. For analytic functions f (z) =

∞ 

an z

n

and g(z) =

n=0

∞ 

bn z n (z ∈ U) ,

n=0

by f ∗ g we denote the Hadamard product or convolution of f and g, defined by (f ∗ g) (z) =

∞ 

an bn z n .

n=0

Received October 8, 2007, accepted February 21, 2008. Communicated by H. M. Srivastava. 2000 Mathematics Subject Classification: 30C45, 26A33. Key words and phrases: Dziok–Srivastava operator, Bernardi operator, Hadamard product (or convolution), Subordination, Starlike functions, Convex functions.

2001

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Jacek Dziok and Janusz Sok o´ l/

Moreover, we say that a function f is subordinate to a function g, and write f (z) ≺ g(z), if and only if there exists a function ω, analytic in U such that ω(0) = 0, |ω(z)| < 1

(z ∈U) ,

and f (z) = g(ω(z)) (z ∈ U) . In particular, if g is univalent in U , we have the following equivalence f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U ).

(2)

Let K denote the class of convex function defined by     zf  (z) > 0, z ∈ U . K := f ∈ A : Re 1 +  f (z) Moreover we recall the class of function introduced by Janowski [6]     zf  (z) 1 + az ∗ 1 + az := f ∈ A : ≺ , z ∈ U (−1 ≤ b < a ≤ 1) . (3) S 1 + bz f (z) 1 + bz  In particular we have the class of starlike functions S ∗ := S ∗ 1+z 1−z . In this paper  we take advantage of S ∗ 1+az 1+bz to define other class of functions. Let q, s ∈ N = {1, 2, ...}, q ≤ s + 1. For complex parameters a1 , . . ., aq and b1 , . . . , bs, (bj = 0, −1, −2, . . .; j = 1, . . . , s), the generalized hypergeometric function q Fs (a1 , . . . , aq ; b1 , . . ., bs ; z) is defined by q Fs (a1 , . . . , aq ; b1 , . . ., bs ; z)

=

∞  (a1 )n · . . . · (aq )n z n n=0

(b1 )n · . . . · (bs)n n!

where (λ)n is the Pochhammer symbol defined by

1 (λ)n = λ(λ + 1) · . . . · (λ + n − 1)

(z ∈ U),

(n = 0) (n ∈ N) .

Let us consider the Dziok–Srivastava operator [4] (see also [3] and [5]) H:A→A such that Hf (z) = H(a1 , . . . , aq ; b1 , . . . , bs)f (z) = {z · q Fs (a1 , . . . , aq ; b1, . . . , bs; z)} ∗f (z).

Some Inclusion Properties of Certain Class of Analytic Functions

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We observe that for a function f of the form (1), we have H(a1 , . . ., aq ; b1, . . . , bs)f (z) = z +

(4)

∞ 

An an z n ,

n=2

where An =

(5)

(a1 )n−1 · . . . · (aq )n−1 . (b1 )n−1 · . . . · (bs)n−1 · (n − 1)!

The Dziok-Srivastava operator H(a1 , . . ., aq ; b1, . . . , bs) includes various other linear operators which were considered in earlier works (see [11], [12] and [13]). In particular we recall the Bernardi integral operator [1] Jν : A → A, defined by (6)

ν +1 Jν [f (z)] = zν

z

tν−1 f (t)dt

(ν ∈ C ).

0

For f ∈ A of the form (1) we have (7)

Jν [f (z)] = z +

∞  ν +1 an z n . ν+n n=2

The Bernardi operator and the Dziok–Srivastava operator are connected in the following way Jν [f (z)] = H(1 + ν, 1; ν + 2)f (z). Let suppose (8)

−1 ≤ B ≤ 0 and |A| < 1 (A ∈ C) .

We denote by V (q, s; A, B) the class of functions f of the form (1) which satisfy the following condition: 1 + Az z [Hf (z)]  ≺ (z ∈ U) . Hf (z) 1 + Bz

1+Az > 0 for z ∈ U, Thus By (8) we have Re 1+Bz

(9)

(10)

f ∈ V (q, s; A, B) ⇒ Hf (z) ∈ S ∗ .

Moreover  for −1 ≤ B < A ≤ 1 this means that Hf (z) belongs to the class 1+Az ∗ S 1+Bz defined by (3). After some calculations we obtain (11)

ai H(ai + 1)f (z) = zH f (z) + (ai − 1)Hf (z),

i = 1, ..., q,

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where, for convenience, H(ai + m)f (z) = H(a1 , . . . , ai + m, . . ., aq ; b1, . . . , bs)f (z),

i = 1, ..., q.

By (11) the condition (9) is for each ai , i = 1, ..., q equivalent the following subordination 1 + Az H(ai + 1)f (z) + 1 − ai ≺ (z ∈ U) . (12) ai Hf (z) 1 + Bz Therefore we use following alternatively notation V (q, s; A, B) = V (ai ; A, B). Dziok and Srivastava [4] making use of the generalized hypergeometric function, have introduced a class of analytic functions with negative coefficients. They considered the class V (q, s; A, B) defined by condition (12) where parameters a1 , . . . , aq , b1 , . . . , bs are positive real and -1≤ A < B ≤ 1 . Some inclusion for this class was given in [2]. The main object of this paper is to investigate a inclusion properties of the classes V (q, s; A, B). 2. MAIN RESULTS We begin with a lemma, which will be useful later on. Lemma 1. [8]. Let ν, A ∈ C and B ∈ [−1; 0] satisfy either   (13) Re 1 + AB + ν(1 + B 2 ) ≥| A + B + B(ν + ν¯) | f or B ∈ (−1; 0], or (14)

1 + A > 0 and Re[1 − A + 2ν] ≥ 0 f or B = −1

If f ∈ A and F (z) = Jν [f (z)] is given by (6), then F ∈ A and 1 + Az zF  (z) 1 + Az zf  (z) ≺ ⇒ ≺ . f (z) 1 + Bz F (z) 1 + Bz Lemma 1 in the more general case is in [8], p. 111. Lemma 2. If the function f is of the form (1), then (15)

Hf (z) = Jai −1 [H(ai + 1)f (z)] (i = 1, 2, ..., q),

where Jai −1 is the Bernardi operator (6).

2005

Some Inclusion Properties of Certain Class of Analytic Functions

Proof. From (4) and from (5) we have Hf (z) = z + = z+ 

∞  n=2 ∞ 

(a1 )n−1 · . . . · (aq )n−1 an z n (b1 )n−1 · . . . · (bs)n−1 · (n − 1)! (a1 )n−1 · . . . ·

ai ai +n−1

· (ai + 1)n−1 · . . . · (aq )n−1

(b1 )n−1 · . . . · (bs )n−1 · (n − 1)! 

n=2 ∞ 

an z n

ai z n ∗ [H(ai + 1)f (z)] ai + n − 1 n=1  ∞  (ai − 1) + 1 z n ∗ [H(ai + 1)f (z)] . = (a − 1) + n i n=1 =

Thus by (7) with ν = ai + 1 we obtain (15). Theorem 1. If m ∈ N and i ∈ {1, ..., q}, then (16)

V (ai + m; A, B) ⊆ V (ai ; A, B),

whenever A, B satisfy either (13) or (14) with ν = a i − 1. Proof. It is clear that it is sufficient to prove (16) only for m = 1. Let f ∈ V (ai + 1; A, B), then from (9) we have 1 + Az z[H(ai + 1)f (z)] ≺ H(ai + 1)f (z) 1 + Bz

(z ∈ U) .

Applying Lemma 1 and Lemma 2, by (9) we obtain that f ∈ V (ai ; A, B). It is natural to ask about the inclusion relation (16) when m is not positive integer. Using a different method we will give a partial answer to this question. We will need the following lemma. Lemma 3. [10]. If f ∈ K, g ∈ S ∗ , then for each analytic function h in U, (f ∗ hg)(U) ⊆ coh(U ), (f ∗ g)(U) where coh(U ) denotes the closed convex hull of h(U). Theorem 2. If G(z) =

∞  n=0

(ai )n n+1 (˜ ai )n z

∈ K, then V (˜ ai ; A, B) ⊂V (ai ; A, B).

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Proof. Let f ∈ V (˜ ai; A, B). By the definition of the subordination we have (17)

1 + Aω(z) z[H(˜ ai )f (z)] = := φ[ω(z)] (z ∈ U), H(˜ ai )f (z) 1 + Bω(z)

where φ is convex univalent mapping of U and |ω(z)| < 1 in U with ω(0) = 0 = φ(0) − 1. Moreover, Re[φ(z)] > 0, z ∈U. Applying (17) and the properties of convolution we get 

(18)

z[H(ai )f (z)] = H(ai )f (z)

z

∞  n=0 ∞ 

n=0

=

(ai )n n+1 (˜ ai )n z

∗ H(˜ ai )f (z)

(ai)n n+1 z (˜ ai )n

∗ H(˜ ai )f (z)



G(z) ∗ φ[ω(z)]H(z) G(z) ∗ zH  (z) = =: g(z). G(z) ∗ H(z) G(z) ∗ H(z)

Because H(z) ∈ S ∗ , G(z) ∈ K and φ is convex univalent, then by Lemma 3 we obtain that for z ∈ U the quantity (18) lies in coφ[ω(U )]. By (2) and from the above-mentioned properties of φ we conclude that g defined by(18) is subordinated 1+Az ⊆ S ∗ and finally f ∈ to φ. Thus, by (9) we have that H(ai )f (z) ∈ S ∗ 1+Bz V (ai ; A, B). Lemma 4. [9]. If either 0 < a ≤ c and c ≥ 2 when a, c are real number, or Re[a + c] ≥ 3, Re[a] ≤ Re[c] and Im[a] = Im[c] when a, c are complex, then the function ∞  (a)n n+1 z (z ∈ U) f (z) = (c)n n=0

belongs to the class K of convex functions. Lemma 4 is a special case of Theorem 2.12 or Theorem 2.13 contained in [9]. Theorem 3. Let i ∈ {1, 2, ..., q} . If ai , ˜ai are real number such that 0 < ai ≤ ˜ai and ˜ai ≥ 2 ai are complex number such that or ai , ˜ Re[ai + ˜ai ] ≥ 3 , Re[ai ] ≤ Re[˜ai ] and Im[ai ] = Im[˜ai ], then V (˜ ai ; A, B) ⊆ V (ai ; A, B).

Some Inclusion Properties of Certain Class of Analytic Functions

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Proof. Since H(˜ ai )f (z) ∈ S ∗ , by Lemma 4 the function G(z) =

∞  (ai )n n=0

(˜ ai )n

z n+1 (z ∈ U)

belongs to the class of convex functions K. Using Theorem 1 we obtain that f ∈ V (ai; A, B). Lemma 5. ([8], p.240). If a, b, c are real and satisfy −2 ≤ a < 0, b = 0, −1 ≤ b and c > M (a, b), where M (a, b) = max{2 + |a + b|, 1 − ab}, then the Gaussian hypergeometric function 2 F1 (a, b, c; z) =

∞  (a)n (b)n n=0

(c)n n!

zn

is convex in U. ˜i > 3 + |ai |, then Lemma 6. Let −1 ≤ ai < 1, i ∈ {1, ..., q}. If a ∞  (ai )n n=0

(˜ ai )n

z n+1 ∈ K.

˜i − 1 in Lemma 5. Then we Proof. Let we chose b = 1, a = ai − 1, c = a obtain that the function ∞  (ai − 1)n n z F (z) = (˜ ai − 1)n n=0

is convex for −2 ≤ ai − 1 < 0 and a ˜i − 1 > M (a, b) = 2 + |ai |. It is clear that [F (z) − 1] ∈ K. After some calculations we obtain that G(z) = aa˜ii −1 −1 G(z) =

∞  (ai )n

(˜ ai )n n=0

z n+1

and this ends the proof. Theorem 4. Let −1 ≤ ai < 1, i ∈ {1, ..., q}. If ˜ai > 3 + |ai|, then V (˜ ai ; A, B) ⊆ V (ai; A, B). Proof. The proof runs as the proof of Theorem 3 by using Lemma 6.

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Theorem 5. Let m ∈ N, i ∈ {1, ..., q}. If Re ai > 1, then (19)

V (ai + m; A, B) ⊆ V (ai ; A, B).

Proof. It is clear that it is sufficient to prove (19) only for m = 1. If f ∈ 1+Az ⊆ S ∗. V (ai + 1; A, B), then by (10) we have H(z) :=H(ai + 1)f (z) ∈ S ∗ 1+Bz Let us denote 1 + Aω(z) zH  (z) = := φ[ω(z)] (z ∈ U), H(z) 1 + Bω(z) where φ is convex univalent and |ω(z)| < 1 in U with ω(0) = 0 = φ(0) − 1 Moreover Re[φ(z)] > 0. If Re ai > 1, then the function G(z) =

∞  (ai − 1) + 1 n z (z ∈ U) (ai − 1) + n

n=1

belongs to the class of convex functions K, (Ruscheweych,[9]). Recall that f (z) ∗ G(z) = J1,ai−1 [f (z)] , where Jai −1 is the Bernardi operator defined by (6). From the proof of Lemma 2 we have H(ai )f (z) = G(z) ∗ H(ai + 1)f (z). Thus

[G(z) ∗ zH(z)] G(z) ∗ zH  (z) z[H(ai )f (z)] = = H(ai )f (z) G(z) ∗ H(z) G(z) ∗ H(z) G(z) ∗ φ[ω(z)]H(z) ∈ coφ(U ). = G(z) ∗ H(z)

For the same reasons as in the proof of Theorem 2 we obtain that f ∈ V (ai ; A, B). REFERENCES 1. S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), 429-446. 2. J. Dziok, On some applications of the Briot–Bouquet differential subordinations, J. Math. Anal. Appl., (2007), 295-301. 3. J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct., 14 (2003), 7-18. 4. J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1-13.

Some Inclusion Properties of Certain Class of Analytic Functions

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5. J. Dziok and H. M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Advanced Studies in Contemporary Mathematics, 5(2) (2002), 115-125. 6. W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math., 23 (1970), 159-177. 7. Y. C. Kim and H. M. Srivastava, Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex Variables Theory Appl., 34 (1997), 293-312. 8. S. S. Miller and P. T. Mocanu, Differential subordinations: theory and applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York · Basel, 2000. 9. St. Ruscheweyh, Convolutions in Geometric Function Theory, Sem. Math. Sup., 83, Presses Univ. Montreal, 1982. 10. St. Ruscheweyh and T. Sheil-Small, Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comm. Math. Helv., 48 (1973), 119-135. 11. H. M. Srivastava and M. K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. I and II, J. Math. Anal. Appl., 171 (1992), 1-13; ibid. 192 (1995), 673-688. 12. H. M. Srivastava and S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math. J., 106 (1987), 1-28. 13. H. M. Srivastava and S. Owa (eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992. Jacek Dziok Institute of Mathematics, University of Rzeszo´ w, ul. Rejtana 16A, PL-35-310 Rzeszo´ w, Poland E-mail: [email protected] Janusz Soko´ l/ Department of Mathematics, Rzeszo´ w University of Technology, ul. W. Pola 2, PL-35-959 Rzeszo´ w, Poland E-mail: [email protected]