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.
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[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
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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:
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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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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
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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:
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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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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.
