- Email: [email protected]
The Hardy space for a certain subclass of Bazilevič functions
Applied Mathematics and Computation 183 (2006) 1201–1207 www.elsevier.com/locate/amc
The Hardy space for a certain subclass of Bazilevicˇ functions Yong Chan Kim a, H.M. Srivastava a
b,*
Department of Mathematics Education, Yeungnam University, 214-1 Daedong, Gyongsan 712-749, South Korea b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada
Abstract In the present sequel to a recent paper by Kim and Sugawa [Y.C. Kim, T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinburgh Math. Soc. (Ser. 2) 49 (2006) 131–143], the authors introduce and investigate a general class of Bazilevicˇ functions, which is defined by using the principle of subordination between analytic functions. They also determine the Hardy space to which this subclass of Bazilevicˇ functions belongs under various parametric constraints. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Hardy space; Bazilevicˇ functions; Close-to-convex functions; Starlike functions; Convex functions; Subordination between analytic functions
1. Introduction and definitions Let A denote the class of functions f normalized by 1 X f ðzÞ ¼ z þ an z n ; n¼2
which are analytic in the open unit disk D ¼ fz : z 2 C and jzj < 1g: Also let S, S and K denote the subclasses of A consisting of functions which are, respectively, univalent, starlike and convex in D (see, for details, [3]). If g(z) is starlike in D, P(z) is analytic with RfP ðzÞg > 0
ðz 2 DÞ;
b is any real number and a > 0, then the function 1=ðaþibÞ Z z a P ðfÞfgðfÞg fib1 df f ðzÞ ¼ ða þ ibÞ 0
*
Corresponding author. E-mail addresses: [email protected] (Y.C. Kim), [email protected] (H.M. Srivastava).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.044
ð1:1Þ
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was shown by Bazilevicˇ [1] (see also Pommerenke [11]) to be an analytic and univalent function in D. The power functions appearing in the formula (1.1) and elsewhere in this investigation are meant as principal values. We shall call a function satisfying (1.1) a Bazilevicˇ function. Let us denote the class of such functions by BðP ; g; a; bÞ. In particular, if we put b = 0 in (1.1), a function f(z) belonging to the class BðP ; g; a; 0Þ :¼ BðaÞ is said to be a Bazilevicˇ function of type a. Sheil-Small [12] investigated the geometric meaning for the class BðP ; g; a; bÞ of Bazilevicˇ functions (see also [14,15] for further information on Bazilevicˇ functions). For analytic functions g and h in D, g is said to be subordinate to h if there exists an analytic function x such that xð0Þ ¼ 0;
jxðzÞj < 1
and
gðzÞ ¼ hðxðzÞÞ ðz 2 DÞ:
This subordination will be denoted by g h or, conventionally, by gðzÞ hðzÞ: In particular, when h is univalent in D, g h if and only if gð0Þ ¼ hð0Þ and
gðDÞ hðDÞ:
Let M be the class of zero-free analytic functions u in D with the normalization condition u(0) = 1. Then, following the earlier work by Ma and Minda [8], Kim and Sugawa [7] introduced the subclasses S ðuÞ and KðuÞ of A as the sets of functions f 2 A satisfying, respectively, the following subordination conditions: zf 0 ðzÞ uðzÞ f ðzÞ and 1þ
zf 00 ðzÞ uðzÞ; f 0 ðzÞ
for each u 2 M. By definition, it is clear that f 2 KðuÞ () zf 0 2 S ðuÞ:
ð1:2Þ
We note also that S ðuÞ S ðwÞ
and
KðuÞ KðwÞ
ðu wÞ:
Next, for A and B such that 1 5 B < A 5 1, let us define the Mo¨bius transformation uA,B by 1 þ Az : uA;B ðzÞ ¼ 1 þ Bz It is fairly obvious that the Mo¨bius transformation uA,B is the conformal map of the unit disk D onto the disk symmetrical with respect to the real axis having the center at 1 AB 1 B2 and the radius equal to AB : 1 B2 The corresponding classes KðuA;B Þ and S ðuA;B Þ were investigated by (for example) Janowski [4,5] and Silverman and Silvia [13]. We observe also that S ¼ S ðu1;1 Þ
and
K ¼ Kðu1;1 Þ:
For functions u, w 2 M, following [7], we denote by Cðu; wÞ the set of all functions f in A for which there exists a function h 2 KðuÞ such that f0 w: h0
Y.C. Kim, H.M. Srivastava / Applied Mathematics and Computation 183 (2006) 1201–1207
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A function f 2 A is said to be close-to-convex if there exist a convex function h 2 K ¼ Kðu1;1 Þ and a real constant c (jcj < p/2) such that 0 ic f ðzÞ R e > 0 ðz 2 DÞ: h0 ðzÞ The last condition is equivalent to the following subordination: f0 wc ; h0 where wc ðzÞ ¼
1 þ eic z : 1 eic z
Therefore, the class of close-to-convex functions can be described as the union of the function classes Cðu1;1 ; wc Þ over p/2 < c < p/2. In an analogous way, for functions u, w 2 M, we denote by Bu;w ðP ; g; a; bÞ the class of functions f 2 A which satisfy (1.1), where g 2 S ðuÞ, P(z) is analytic with P(z) w(z) in D, b is real and a > 0. In particular, if we put uðzÞ ¼ wðzÞ ¼ u1;1 ðzÞ ¼
1þz ; 1z
we easily see that Bu1;1 ;u1;1 ðP ; g; a; bÞ ¼ BðP ; g; a; bÞ: Also, by a simple calculation, we get P ðzÞ ¼
zibþ1 f 0 ðzÞff ðzÞga1þib : a fgðzÞg
Hence it follows from (1.2) that Bu;w ðP ; g; 1; 0Þ ¼ Cðu; wÞ: Furthermore, by taking w(z) = u1,1(z), it is easily verified that Bu;u1;1 ð1; g; 1; 0Þ ¼ KðuÞ and Bu;u1;1
0 zg ; g; 1; 0 ¼ S ðuÞ: g
Even though the class BðP ; g; a; bÞ is a subclass of the class S of univalent functions, we remark that the class Bu;w ðP ; g; a; bÞ does not need to be in the class S of univalent functions. But, if u and w have positive real parts, then the class Bu;w ðP ; g; a; bÞ is a subclass of Bazilevicˇ functions. For example, by taking b ¼ 0;
uðzÞ ¼ uC;D ðzÞ
and
wðzÞ ¼ uA;B ðzÞ:
Noor [10] defined a subclass Ba ½A; B; C; D of Bazilevicˇ functions, that is, BuC;D ;uA;B ðP ; g; a; 0Þ ¼: Ba ½A; B; C; D: Throughout this paper, for the sake of brevity, we shall make use of the simplified notation BuA;B ;w ðaÞ for the function class BuA;B ;w ðP ; g; a; 0Þ.
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Finally, let Hp (0 < p 5 1) denote the Hardy space of analytic functions f(z) in D, and define the integral means by 8 Z 1=p 2p > > ih p < 1 jf ðre Þj dh ð0 < p < 1Þ 2p 0 ð1:3Þ M p ðr; f Þ ¼ > > : max jf ðzÞj ðp ¼ 1Þ: jzj5r
Then, by definition, an analytic function f(z) in D belongs to the Hardy space Hp (0 < p 5 1) if kf kp :¼ lim M p ðr; f Þ < 1: r!1
For a function f in the class BðaÞ, Miller [9] investigated the Hardy spaces to which f(z) and f 0 (z) belong. In the present paper, when uðzÞ ¼ uA;B ðzÞ
and
RfwðzÞg > 0
ðz 2 DÞ;
we carry out the corresponding investigations for the function class BuA;B ;w ðaÞ. 2. The Hardy space for the function class BuA;B ;w ðaÞ For u 2 M, we define the functions hu and ku in A by the following relations: zh0u ðzÞ ¼ uðzÞ hu ðzÞ
and
1þ
zk 00u ðzÞ ¼ uðzÞ; k 0u ðzÞ
ð2:1Þ
that is, hu ðzÞ ¼ z exp
Z
z 0
uðtÞ 1 dt t
and
k u ðzÞ ¼
Z
Z
z
f
exp 0
For example, if 1 6 B < A 6 1, it is well known that ( ðABÞ=B ðB 6¼ 0Þ; zð1 þ BzÞ huA;B ðzÞ ¼ zk 0uA;B ðzÞ ¼ Az ze ðB ¼ 0Þ;
0
uðtÞ 1 dt df: t
ð2:2Þ
and 8 A=B 1 > < A fð1 þ BzÞ 1g k uA;B ðzÞ ¼ B1 logð1 þ BzÞ > : 1 Az ðe 1Þ A
ðA 6¼ 0; B 6¼ 0Þ; ðA ¼ 0Þ; ðB ¼ 0Þ:
We shall need the following lemma in our investigation. Lemma [7, Lemma 3.1]. Suppose that a function u 2 M is univalent in D and uðDÞ is starlike with respect to 1. Then the subordination f 0 k 0u holds true for every f 2 KðuÞ. Our main result is stated as the following theorem. Theorem. Let a > 0 and let 1 5 B < A 5 1. Suppose that w 2 M \ Hp has positive real part in D and ap < 1. If 1 < B or A < 0, then ap (i) BuA;B ;w ðaÞ H1þapp for 0 < p < 1 þ ap 0 < a < 1p ; 0 < a < 1p ; a 6¼ 1 . (ii) BuA;B ;w ðaÞ H1 for p=1 þ ap Proof. We shall omit some of the details of our proof here, because there are many similarities between this proof and the proof of Theorem 1 in [9] (see also Theorem 1 of [6]). Let f 2 BuA;B ;w ðaÞ. If we set
Y.C. Kim, H.M. Srivastava / Applied Mathematics and Computation 183 (2006) 1201–1207
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a f ðzÞ F ðzÞ ¼ ; z then we find from [9, Eq. (3)] that Z 2p Z 2p Z 2p aF ðzÞk afgðzÞga P ðzÞk k 0 jF ðzÞj dh5 z dh ¼: I 1 ðrÞ þ I 2 ðrÞ: dh þ zaþ1 0 0 0
ð2:3Þ
Since RfuA;B ðzÞg > 0 and RfwðzÞg > 0
ðz 2 DÞ;
we have BuA;B ;w ðaÞ BðaÞ S: Hence, by the known result [2, Theorem 3.16], limr!1I2(r) exists if 1 ak < : 2
ð2:4Þ
In order to investigate I1(r), by Ho¨lder’s inequality, we find that Z 2p q10 Z 2p p10 Z 2p 1 1 ak k akq0 kp0 jgðzÞj jP ðzÞj dh5 jgðzÞj dh jP ðzÞj dh þ ¼ 1 ¼: J 1 ðrÞ J 2 ðrÞ: p 0 q0 0 0 0
ð2:5Þ
Since P w
and
w 2 Hp ;
by the Littlewood Subordination Theorem [2, Theorem 1.7], we get P ðzÞ 2 Hp : This implies that limr!1J2(r) exists if kp0 5p: Now, if we put u = uA,B, we can rewrite the equivalence relationship (1.2) as follows: Z z gðtÞ dt 2 KðuA;B Þ: g 2 S ðuA;B Þ () GðzÞ ¼ t 0
ð2:6Þ
ð2:7Þ
Since the function uA,B is univalent 2 D and uA;B ðDÞ is starlike with respect to 1, it follows from the above Lemma that gðzÞ k 0uA;B : z In view of Eq. (2.2), if 1 < B < 1, then k 0uA;B is a bounded analytic function in D. Also, if A < 0, then 2 H1 . From these facts, we see that k 0uA;B 2 H1 if and only if 1 < B or A < 0. Hence k 0uA;B 2 H1 , so that gðzÞ 2 H1 : This implies that limr!1J1(r) exists if
k 0uA;B
kaq0 51:
ð2:8Þ
Since 1 1 þ ¼ 1; p 0 q0 by making use of (2.6) and (2.8), we obtain p k5 : 1 þ ap
ð2:9Þ
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Moreover, if ap < 1, the inequality (2.4) holds true. Hence, from (2.3) and (2.5), it is easy to see that p
F 0 ðzÞ 2 H1þap : If 0 < p < 1 + ap, it follows from the Hardy-Littlewood Theorem [2, Theorem 5.12] that p
F ðzÞ 2 H1þapp ; which implies that ap
f ðzÞ 2 H1þapp : If p = 1 + ap, then F 0 ðzÞ 2 H1 . Hence, from the known result [2, Theorem 3.11], we have F ðzÞ 2 H1 , so that f ðzÞ 2 H1 : This completes the proof of the Theorem.
h
3. An interesting consequence of the Theorem By setting a = 1 in our Theorem, and then combining the resulting assertion of the Theorem with another known result [2, Theorem 3.2], we arrive at the following result. Corollary. Let A and B be as assumed in the Theorem. If w has positive real part in D with w(0) = 1, then \ CðuA;B ; wÞ Hp : ð3:1Þ 0
In particular, \ 1þz C uA;B ; Hp : 1z 0
ð3:2Þ
Remark. For A and B as assumed in the Theorem, Kim and Sugawa [7, Corollary 5.2] proved that, if 1 5 p < 1 and w 2 M \ Hp , then CðuA;B ; wÞ Hp :
ð3:3Þ
Acknowledgements The present investigation was completed during the first-named author’s visit to the University of Victoria from March 2006 to August 2006 while he was on Study Leave from Yeungnam University at Gyongsan. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] I.E. Bazilevicˇ, On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb. 37 (79) (1955) 471–476. [2] P.L. Duren, Theory of Hp Spaces, A Series of Comprehensive Studies in Mathematics, vol. 38, Academic Press, New York and London, 1970. [3] P.L. Duren, Univalent Functions, A Series of Comprehensive Studies in Mathematics, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983. [4] W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 23 (1973) 159–177. [5] W. Janowski, Some extremal problems for certain families of analytic functions. II, Bull. Acad. Polon. Sci. Se´r. Sci. Math. Astronom. Phys. 21 (1973) 17–25. [6] Y.C. Kim, K.S. Lee, H.M. Srivastava, Certain classes of integral operators associated with the Hardy space of analytic functions, Complex Variables Theory Appl. 20 (1992) 179–200. [7] Y.C. Kim, T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinburgh Math. Soc. (Ser. 2) 49 (2006) 131–143.
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[8] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Z. Li, F. Ren, L. Yang, S. Zhang (Eds.), Proceedings of the Conference on Complex Analysis, International Press, Cambridge, Massachusetts, 1992, pp. 157–169. [9] S.S. Miller, The Hardy class of a Bazilevicˇ function and its derivative, Proc. Amer. Math. Soc. 30 (1971) 225–239. [10] K.I. Noor, On some univalent integral operators, J. Math. Anal. Appl. 128 (1987) 586–592. ¨ ber die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965) 159–173. [11] C. Pommerenke, U [12] T. Sheil-Small, On Bazilevicˇ functions, Quart. J. Math. Oxford (Ser. 2) 23 (1972) 135–142. [13] H. Silverman, E.M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math. 37 (1985) 48–61. [14] R. Singh, On Bazilevicˇ functions, Proc. Amer. Math. Soc. 38 (1973) 261–271. [15] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.
The Hardy space for a certain subclass of Bazilevicˇ functions Yong Chan Kim a, H.M. Srivastava a
b,*
Department of Mathematics Education, Yeungnam University, 214-1 Daedong, Gyongsan 712-749, South Korea b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada
Abstract In the present sequel to a recent paper by Kim and Sugawa [Y.C. Kim, T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinburgh Math. Soc. (Ser. 2) 49 (2006) 131–143], the authors introduce and investigate a general class of Bazilevicˇ functions, which is defined by using the principle of subordination between analytic functions. They also determine the Hardy space to which this subclass of Bazilevicˇ functions belongs under various parametric constraints. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Hardy space; Bazilevicˇ functions; Close-to-convex functions; Starlike functions; Convex functions; Subordination between analytic functions
1. Introduction and definitions Let A denote the class of functions f normalized by 1 X f ðzÞ ¼ z þ an z n ; n¼2
which are analytic in the open unit disk D ¼ fz : z 2 C and jzj < 1g: Also let S, S and K denote the subclasses of A consisting of functions which are, respectively, univalent, starlike and convex in D (see, for details, [3]). If g(z) is starlike in D, P(z) is analytic with RfP ðzÞg > 0
ðz 2 DÞ;
b is any real number and a > 0, then the function 1=ðaþibÞ Z z a P ðfÞfgðfÞg fib1 df f ðzÞ ¼ ða þ ibÞ 0
*
Corresponding author. E-mail addresses: [email protected] (Y.C. Kim), [email protected] (H.M. Srivastava).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.044
ð1:1Þ
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was shown by Bazilevicˇ [1] (see also Pommerenke [11]) to be an analytic and univalent function in D. The power functions appearing in the formula (1.1) and elsewhere in this investigation are meant as principal values. We shall call a function satisfying (1.1) a Bazilevicˇ function. Let us denote the class of such functions by BðP ; g; a; bÞ. In particular, if we put b = 0 in (1.1), a function f(z) belonging to the class BðP ; g; a; 0Þ :¼ BðaÞ is said to be a Bazilevicˇ function of type a. Sheil-Small [12] investigated the geometric meaning for the class BðP ; g; a; bÞ of Bazilevicˇ functions (see also [14,15] for further information on Bazilevicˇ functions). For analytic functions g and h in D, g is said to be subordinate to h if there exists an analytic function x such that xð0Þ ¼ 0;
jxðzÞj < 1
and
gðzÞ ¼ hðxðzÞÞ ðz 2 DÞ:
This subordination will be denoted by g h or, conventionally, by gðzÞ hðzÞ: In particular, when h is univalent in D, g h if and only if gð0Þ ¼ hð0Þ and
gðDÞ hðDÞ:
Let M be the class of zero-free analytic functions u in D with the normalization condition u(0) = 1. Then, following the earlier work by Ma and Minda [8], Kim and Sugawa [7] introduced the subclasses S ðuÞ and KðuÞ of A as the sets of functions f 2 A satisfying, respectively, the following subordination conditions: zf 0 ðzÞ uðzÞ f ðzÞ and 1þ
zf 00 ðzÞ uðzÞ; f 0 ðzÞ
for each u 2 M. By definition, it is clear that f 2 KðuÞ () zf 0 2 S ðuÞ:
ð1:2Þ
We note also that S ðuÞ S ðwÞ
and
KðuÞ KðwÞ
ðu wÞ:
Next, for A and B such that 1 5 B < A 5 1, let us define the Mo¨bius transformation uA,B by 1 þ Az : uA;B ðzÞ ¼ 1 þ Bz It is fairly obvious that the Mo¨bius transformation uA,B is the conformal map of the unit disk D onto the disk symmetrical with respect to the real axis having the center at 1 AB 1 B2 and the radius equal to AB : 1 B2 The corresponding classes KðuA;B Þ and S ðuA;B Þ were investigated by (for example) Janowski [4,5] and Silverman and Silvia [13]. We observe also that S ¼ S ðu1;1 Þ
and
K ¼ Kðu1;1 Þ:
For functions u, w 2 M, following [7], we denote by Cðu; wÞ the set of all functions f in A for which there exists a function h 2 KðuÞ such that f0 w: h0
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A function f 2 A is said to be close-to-convex if there exist a convex function h 2 K ¼ Kðu1;1 Þ and a real constant c (jcj < p/2) such that 0 ic f ðzÞ R e > 0 ðz 2 DÞ: h0 ðzÞ The last condition is equivalent to the following subordination: f0 wc ; h0 where wc ðzÞ ¼
1 þ eic z : 1 eic z
Therefore, the class of close-to-convex functions can be described as the union of the function classes Cðu1;1 ; wc Þ over p/2 < c < p/2. In an analogous way, for functions u, w 2 M, we denote by Bu;w ðP ; g; a; bÞ the class of functions f 2 A which satisfy (1.1), where g 2 S ðuÞ, P(z) is analytic with P(z) w(z) in D, b is real and a > 0. In particular, if we put uðzÞ ¼ wðzÞ ¼ u1;1 ðzÞ ¼
1þz ; 1z
we easily see that Bu1;1 ;u1;1 ðP ; g; a; bÞ ¼ BðP ; g; a; bÞ: Also, by a simple calculation, we get P ðzÞ ¼
zibþ1 f 0 ðzÞff ðzÞga1þib : a fgðzÞg
Hence it follows from (1.2) that Bu;w ðP ; g; 1; 0Þ ¼ Cðu; wÞ: Furthermore, by taking w(z) = u1,1(z), it is easily verified that Bu;u1;1 ð1; g; 1; 0Þ ¼ KðuÞ and Bu;u1;1
0 zg ; g; 1; 0 ¼ S ðuÞ: g
Even though the class BðP ; g; a; bÞ is a subclass of the class S of univalent functions, we remark that the class Bu;w ðP ; g; a; bÞ does not need to be in the class S of univalent functions. But, if u and w have positive real parts, then the class Bu;w ðP ; g; a; bÞ is a subclass of Bazilevicˇ functions. For example, by taking b ¼ 0;
uðzÞ ¼ uC;D ðzÞ
and
wðzÞ ¼ uA;B ðzÞ:
Noor [10] defined a subclass Ba ½A; B; C; D of Bazilevicˇ functions, that is, BuC;D ;uA;B ðP ; g; a; 0Þ ¼: Ba ½A; B; C; D: Throughout this paper, for the sake of brevity, we shall make use of the simplified notation BuA;B ;w ðaÞ for the function class BuA;B ;w ðP ; g; a; 0Þ.
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Finally, let Hp (0 < p 5 1) denote the Hardy space of analytic functions f(z) in D, and define the integral means by 8 Z 1=p 2p > > ih p < 1 jf ðre Þj dh ð0 < p < 1Þ 2p 0 ð1:3Þ M p ðr; f Þ ¼ > > : max jf ðzÞj ðp ¼ 1Þ: jzj5r
Then, by definition, an analytic function f(z) in D belongs to the Hardy space Hp (0 < p 5 1) if kf kp :¼ lim M p ðr; f Þ < 1: r!1
For a function f in the class BðaÞ, Miller [9] investigated the Hardy spaces to which f(z) and f 0 (z) belong. In the present paper, when uðzÞ ¼ uA;B ðzÞ
and
RfwðzÞg > 0
ðz 2 DÞ;
we carry out the corresponding investigations for the function class BuA;B ;w ðaÞ. 2. The Hardy space for the function class BuA;B ;w ðaÞ For u 2 M, we define the functions hu and ku in A by the following relations: zh0u ðzÞ ¼ uðzÞ hu ðzÞ
and
1þ
zk 00u ðzÞ ¼ uðzÞ; k 0u ðzÞ
ð2:1Þ
that is, hu ðzÞ ¼ z exp
Z
z 0
uðtÞ 1 dt t
and
k u ðzÞ ¼
Z
Z
z
f
exp 0
For example, if 1 6 B < A 6 1, it is well known that ( ðABÞ=B ðB 6¼ 0Þ; zð1 þ BzÞ huA;B ðzÞ ¼ zk 0uA;B ðzÞ ¼ Az ze ðB ¼ 0Þ;
0
uðtÞ 1 dt df: t
ð2:2Þ
and 8 A=B 1 > < A fð1 þ BzÞ 1g k uA;B ðzÞ ¼ B1 logð1 þ BzÞ > : 1 Az ðe 1Þ A
ðA 6¼ 0; B 6¼ 0Þ; ðA ¼ 0Þ; ðB ¼ 0Þ:
We shall need the following lemma in our investigation. Lemma [7, Lemma 3.1]. Suppose that a function u 2 M is univalent in D and uðDÞ is starlike with respect to 1. Then the subordination f 0 k 0u holds true for every f 2 KðuÞ. Our main result is stated as the following theorem. Theorem. Let a > 0 and let 1 5 B < A 5 1. Suppose that w 2 M \ Hp has positive real part in D and ap < 1. If 1 < B or A < 0, then ap (i) BuA;B ;w ðaÞ H1þapp for 0 < p < 1 þ ap 0 < a < 1p ; 0 < a < 1p ; a 6¼ 1 . (ii) BuA;B ;w ðaÞ H1 for p=1 þ ap Proof. We shall omit some of the details of our proof here, because there are many similarities between this proof and the proof of Theorem 1 in [9] (see also Theorem 1 of [6]). Let f 2 BuA;B ;w ðaÞ. If we set
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a f ðzÞ F ðzÞ ¼ ; z then we find from [9, Eq. (3)] that Z 2p Z 2p Z 2p aF ðzÞk afgðzÞga P ðzÞk k 0 jF ðzÞj dh5 z dh ¼: I 1 ðrÞ þ I 2 ðrÞ: dh þ zaþ1 0 0 0
ð2:3Þ
Since RfuA;B ðzÞg > 0 and RfwðzÞg > 0
ðz 2 DÞ;
we have BuA;B ;w ðaÞ BðaÞ S: Hence, by the known result [2, Theorem 3.16], limr!1I2(r) exists if 1 ak < : 2
ð2:4Þ
In order to investigate I1(r), by Ho¨lder’s inequality, we find that Z 2p q10 Z 2p p10 Z 2p 1 1 ak k akq0 kp0 jgðzÞj jP ðzÞj dh5 jgðzÞj dh jP ðzÞj dh þ ¼ 1 ¼: J 1 ðrÞ J 2 ðrÞ: p 0 q0 0 0 0
ð2:5Þ
Since P w
and
w 2 Hp ;
by the Littlewood Subordination Theorem [2, Theorem 1.7], we get P ðzÞ 2 Hp : This implies that limr!1J2(r) exists if kp0 5p: Now, if we put u = uA,B, we can rewrite the equivalence relationship (1.2) as follows: Z z gðtÞ dt 2 KðuA;B Þ: g 2 S ðuA;B Þ () GðzÞ ¼ t 0
ð2:6Þ
ð2:7Þ
Since the function uA,B is univalent 2 D and uA;B ðDÞ is starlike with respect to 1, it follows from the above Lemma that gðzÞ k 0uA;B : z In view of Eq. (2.2), if 1 < B < 1, then k 0uA;B is a bounded analytic function in D. Also, if A < 0, then 2 H1 . From these facts, we see that k 0uA;B 2 H1 if and only if 1 < B or A < 0. Hence k 0uA;B 2 H1 , so that gðzÞ 2 H1 : This implies that limr!1J1(r) exists if
k 0uA;B
kaq0 51:
ð2:8Þ
Since 1 1 þ ¼ 1; p 0 q0 by making use of (2.6) and (2.8), we obtain p k5 : 1 þ ap
ð2:9Þ
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Moreover, if ap < 1, the inequality (2.4) holds true. Hence, from (2.3) and (2.5), it is easy to see that p
F 0 ðzÞ 2 H1þap : If 0 < p < 1 + ap, it follows from the Hardy-Littlewood Theorem [2, Theorem 5.12] that p
F ðzÞ 2 H1þapp ; which implies that ap
f ðzÞ 2 H1þapp : If p = 1 + ap, then F 0 ðzÞ 2 H1 . Hence, from the known result [2, Theorem 3.11], we have F ðzÞ 2 H1 , so that f ðzÞ 2 H1 : This completes the proof of the Theorem.
h
3. An interesting consequence of the Theorem By setting a = 1 in our Theorem, and then combining the resulting assertion of the Theorem with another known result [2, Theorem 3.2], we arrive at the following result. Corollary. Let A and B be as assumed in the Theorem. If w has positive real part in D with w(0) = 1, then \ CðuA;B ; wÞ Hp : ð3:1Þ 0
In particular, \ 1þz C uA;B ; Hp : 1z 0
ð3:2Þ
Remark. For A and B as assumed in the Theorem, Kim and Sugawa [7, Corollary 5.2] proved that, if 1 5 p < 1 and w 2 M \ Hp , then CðuA;B ; wÞ Hp :
ð3:3Þ
Acknowledgements The present investigation was completed during the first-named author’s visit to the University of Victoria from March 2006 to August 2006 while he was on Study Leave from Yeungnam University at Gyongsan. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] I.E. Bazilevicˇ, On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb. 37 (79) (1955) 471–476. [2] P.L. Duren, Theory of Hp Spaces, A Series of Comprehensive Studies in Mathematics, vol. 38, Academic Press, New York and London, 1970. [3] P.L. Duren, Univalent Functions, A Series of Comprehensive Studies in Mathematics, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983. [4] W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 23 (1973) 159–177. [5] W. Janowski, Some extremal problems for certain families of analytic functions. II, Bull. Acad. Polon. Sci. Se´r. Sci. Math. Astronom. Phys. 21 (1973) 17–25. [6] Y.C. Kim, K.S. Lee, H.M. Srivastava, Certain classes of integral operators associated with the Hardy space of analytic functions, Complex Variables Theory Appl. 20 (1992) 179–200. [7] Y.C. Kim, T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinburgh Math. Soc. (Ser. 2) 49 (2006) 131–143.
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[8] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Z. Li, F. Ren, L. Yang, S. Zhang (Eds.), Proceedings of the Conference on Complex Analysis, International Press, Cambridge, Massachusetts, 1992, pp. 157–169. [9] S.S. Miller, The Hardy class of a Bazilevicˇ function and its derivative, Proc. Amer. Math. Soc. 30 (1971) 225–239. [10] K.I. Noor, On some univalent integral operators, J. Math. Anal. Appl. 128 (1987) 586–592. ¨ ber die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965) 159–173. [11] C. Pommerenke, U [12] T. Sheil-Small, On Bazilevicˇ functions, Quart. J. Math. Oxford (Ser. 2) 23 (1972) 135–142. [13] H. Silverman, E.M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math. 37 (1985) 48–61. [14] R. Singh, On Bazilevicˇ functions, Proc. Amer. Math. Soc. 38 (1973) 261–271. [15] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.