On a class of holomorphic functions defined by Salagean differential ...

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Mar 15, 2005 - If f and g are analytic functions in U, then we say that f is subordinate to g, written f 0 g, or f ًzق 0 gًzق, if there is a function w analytic in U with ...
Complex Variables, Vol. 50, No. 4, 15 March 2005, 257–264

On a class of holomorphic functions defined by Salagean differential operator GEORGIA IRINA OROS* Department of Mathematics, University of Oradea, Str. Armatei Romaˆne, No. 5, 410087 Oradea, Romania Communicated by R.P. Gilbert (Received 6 August 2004; in final form 31 August 2004) By using the differential operator Dm f ðzÞ, z 2 U (Definition 1), we introduce a class of holomorphic functions, denoted by Mnm ðÞ, and we obtain some inclusion relations. Keywords: Holomorphic function; Convex function; Integral operator AMS Subject Classification: 30C45

1. Introduction and preliminaries Denote by U the unit disc of the complex plane: U ¼ fz 2 C : jzj < 1g: Let H½U be the space of holomorphic function in U. We let:   An ¼ f 2 H ½U , f ðzÞ ¼ z þ anþ1 znþ1 þ    , z 2 U with A1 ¼ A. We let H½a, n denote the class of analytic functions in U of the form f ðzÞ ¼ a þ an zn þ anþ1 znþ1 þ    , 

Let K¼

z 2 U:

zf 00 ðzÞ þ 1 > 0, z 2 U f 2 A : Re 0 f ðzÞ



denote the class of normalized convex functions in U.

*E-mail: [email protected] Complex Variables ISSN 0278-1077 print: ISSN 1563-5066 online  2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/02781070412331328594

258

G.I. Oros

If f and g are analytic functions in U, then we say that f is subordinate to g, written f  g, or f ðzÞ  gðzÞ, if there is a function w analytic in U with wð0Þ ¼ 0, jwðzÞj < 1, for all z 2 U, such that f ðzÞ ¼ g½wðzÞ for z 2 U. If g is univalent, then f  g if f ð0Þ ¼ gð0Þ and f ðUÞ  gðUÞ. We use the following subordination results. LEMMA A (Miller and Mocanu [1] [1, p. 71]) Let h be a convex function with hð0Þ  a and let  2 fC be a complex number with Re   0. If p 2 H ½U with pð0Þ ¼ a and 1 pðzÞ þ zp0 ðzÞ  hðzÞ  then pðzÞ  qðzÞ  hðzÞ where

qðzÞ ¼



Z

nzð=nÞ1

z

hðtÞtð=nÞ1 dt:

0

The function q is convex and is the best dominant. (The definition of best dominant in given in [2].) LEMMA B (Miller and Mocanu [2])

Let g be a convex function in U and let

hðzÞ ¼ gðzÞ þ nzg0 ðzÞ where >0 and n is a positive integer. If pðzÞ ¼ gð0Þ þ pn zn þ    is holomorphic in U and pðzÞ þ zp0 ðzÞ  hðzÞ then pðzÞ  gðzÞ and this result is sharp. Definition 1 (Sa˘la˘gean [5])

For f 2 A and n  0 we define the operator Dn f by D0 f ðzÞ ¼ f ðzÞ Dnþ1 f ðzÞ ¼ z½Dn f ðzÞ0 ,

z 2 U:

Holomorphic functions defined by Salagean differential operator

259

2. Main results Definition 2 If  : THEOREM 1

ð1Þ

If   which is equivalent to pðzÞ þ zp0 ðzÞ 

1 þ ð2  1Þz  hðzÞ: 1þz

By using Lemma A, we have: pðzÞ  gðzÞ  hðzÞ

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G.I. Oros

where Zz 1 1 þ ð2  1Þt ð1=nÞ1 t gðzÞ ¼ 1=n dt nz 1þt 0  Z z 1 1 tð1=nÞ1 dt 2  1 þ 2ð1  Þ ¼ 1=n nz 1þt 1þt 0 Zz Z 1 2ð1  Þ z tð1=nÞ1 ¼ 1=n ð2  1Þtð1=nÞ1 dt þ dt nz nz1=n 0 1 þ t 0   1 1 1 ¼ 2  1 þ 2ð1  Þ  : n n z1=n The function g is convex and is the best dominant. From pðzÞ  qðzÞ, it results that   1 1 Re pðzÞ > Re gð1Þ ¼ ð, nÞ ¼ 2  1 þ 2ð1  Þ  , n n from which we deduce that Mnmþ1 ðÞ  Mnm ðÞ.

g

For n ¼ 1, this result was obtained in [3]. THEOREM 2

Let g be a convex function, gð0Þ ¼ 1 and let h be a function, such that hðzÞ ¼ gðzÞ þ nzg0 ðzÞ:

If f 2 An and verifies the differential subordination ½Dmþ1 f ðzÞ0  hðzÞ then ½Dm f ðzÞ0  gðzÞ and this result is sharp. Proof

From Dmþ1 f ðzÞ ¼ z½Dm f ðzÞ0 , we obtain ½Dmþ1 f ðzÞ0 ¼ ½Dm f ðzÞ0 þ z½Dm f ðzÞ00 :

If we let pðzÞ ¼ ½Dm f ðzÞ0 , p0 ðzÞ ¼ ½Dm f ðzÞ00 , then we obtain ½Dmþ1 f ðzÞ0 ¼ pðzÞ þ zp0 ðzÞ and (5) becomes pðzÞ þ zp0 ðzÞ  gðzÞ þ nzg0 ðzÞ  hðzÞ:

ð5Þ

Holomorphic functions defined by Salagean differential operator

261

By using Lemma B, we have pðzÞ  gðzÞ,

i.e., ½Dm f ðzÞ0  gðzÞ g

and this result is sharp. For n ¼ 1, this result was obtained in [3]. THEOREM 3

Let h 2 H ½U, with hð0Þ ¼ 1, h0 ð0Þ 6¼ 0, which verifies the inequality   zh00 ðzÞ 1 Re 1 þ 0 > , h ðzÞ 2

z 2 U:

If f 2 An and verifies the differential subordination ½Dmþ1 f ðzÞ0  hðzÞ,

z 2 U,

ð6Þ

then ½Dm f ðzÞ0  gðzÞ where gðzÞ ¼

1 nz1=n

Z

z

hðtÞtð1=nÞ1 dt: 0

The function g is convex and is the best dominant. Proof A simple application of the differential subordination technique [1, Corollary 2.6g.2, p. 66] shows that the function g is convex. From Dmþ1 f ðzÞ ¼ z½Dm f ðzÞ0 we obtain ½Dmþ1 f ðzÞ0 ¼ ½Dm f ðzÞ0 þ z½Dm f ðzÞ00 : If we let pðzÞ ¼ ½Dm f ðzÞ0 , p0 ðzÞ ¼ ½Dm f ðzÞ00 , then we obtain ½Dmþ1 f ðzÞ0 ¼ pðzÞ þ zp0 ðzÞ and (6) becomes pðzÞ þ zp0 ðzÞ  hðzÞ:

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By using Lemma A, we have 1 pðzÞ  gðzÞ ¼ 1=n nz THEOREM 4

Z

z

hðtÞtð1=nÞ1 dt: 0

Let g be a convex function, gð0Þ ¼ 1, and hðzÞ ¼ gðzÞ þ nzg0 ðzÞ:

If f 2 An and verifies the differential subordination ½Dm f ðzÞ0  hðzÞ,

z 2 U,

ð7Þ

then Dm f ðzÞ  gðzÞ, z

z2U

and this result is sharp. Proof

We let pðzÞ ¼ Dm f ðzÞ=z, z 2 U, and we obtain Dm f ðzÞ ¼ zpðzÞ:

By differentiating, we obtain ½Dm f ðzÞ0 ¼ pðzÞ þ zp0 ðzÞ,

z 2 U:

Then (7) becomes pðzÞ þ zp0 ðzÞ  hðzÞ  gðzÞ þ nzg0 ðzÞ By using Lemma B, we have pðzÞ  gðzÞ, i.e. Dm f ðzÞ  gðzÞ: z

g

For n ¼ 1, this result was obtained in [3]. THEOREM 5

Let h 2 HðUÞ, hð0Þ ¼ 1, hð0Þ 6¼ 0 which satisfy the inequality   zh00 ðzÞ 1 Re 1 þ 0 > , h ðzÞ 2

z 2 U:

Holomorphic functions defined by Salagean differential operator

263

If f 2 An and verifies the differential subordination ½Dm f ðzÞ0  hðzÞ,

z 2 U,

ð8Þ

then Dm f ðzÞ  gðzÞ, z where gðzÞ ¼

1 nz1=n

z 2 U,

Z

z 6¼ 0,

z

hðtÞtð1=nÞ1 dt: 0

The function g is convex and is the best dominant. Proof

We let pðzÞ ¼

Dm f ðzÞ , z

z 2 U,

z 6¼ 0:

By differentiating (9) we obtain ½Dm f ðzÞ0 ¼ pðzÞ þ zp0 ðzÞ: Then (8) becomes pðzÞ þ zp0 ðzÞ  hðzÞ,

z 2 U:

By using Lemma A, we have pðzÞ  gðzÞ where gðzÞ ¼

1 nz1=n

Z

z

hðtÞtð1=nÞ1 dt 0

and g is convex and is the best dominant. COROLLARY 1

If f 2 M 2 Mnm ðÞ, then   Dm f ðzÞ 1 1 > 2  1 þ 2ð1  Þ  Re : z n n

Proof

From Theorem 3, we deduce Dm f ðzÞ 1  gðzÞ ¼ 1=n z nz

Z

z

hðtÞtð1=nÞ1 dt, 0

ð9Þ

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G.I. Oros

where hðzÞ ¼

1 þ ð2  1Þz , 1þz

z 2 U:

Hence   Dm f ðzÞ 1 1 > gð1Þ ¼ 2  1 þ 2ð1  Þ  Re : z n n We remark that in the case of meromorphic functions a similar results was obtained by Pap in [4]. References [1] Miller, S.S. and Mocanu, P.T., 2000, Differential Subordinations. Theory and Applications. (New York, Basel: Marcel Dekker Inc.). [2] Miller, S.S. and Mocanu, P.T., 1985, On some classes of first-order differential subordinations. The Michigan Mathematical Journal, 32, 185–195. [3] Georgia Irina Oros, A Class of Holomorphic Functions Defined Using a Differential Operator (to appear). [4] Margit Pap, 1998, On certain subclasses of meromorphic m-valent close-to-convex functions. P.U.M.A., 9(1–2), 155–163. [5] Grigore Stefan Sa˘la˘gean, 1983, Subclasses of univalent functions. Lecture Notes in Mathematics, Vol. 1013 (Berlin: Springer Verlag) pp. 362–372.