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Some properties of a certain class of rational functions
Applied Mathematics and Computation 187 (2007) 403–407 www.elsevier.com/locate/amc
Some properties of a certain class of rational functions R.K. Raina
a,*
, D.K. Bansal
b
a
b
Department of Mathematics, College of Technology and Agricultural Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur 313 001, Rajasthan, India Department of Mathematics, College of Science, Mohan Lal Sukhadia University, Udaipur 313 001, Rajasthan, India
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract In the present paper, we investigate the familiar geometric properties of starlikeness, convexity and spirallikeness of a certain class of rational functions. Relevance with various known (and new) results are also mentioned. Ó 2006 Published by Elsevier Inc. Keywords: Analytic functions; Starlike functions; Convex functions; Spirallike functions
1. Introduction Let A be the class of functions f(z) defined by 1 X ak z k ; f ðzÞ ¼ z þ
ð1:1Þ
k¼2
which are analytic in the open unit disk U ¼ fz : z 2 C; jzj < 1g. The subclasses of the class A denoted by S ðaÞ; KðaÞ and Sp(l, a) are, respectively, the well known subclasses of starlike functions of order a(0 5 a < 1) in U, the convex functions of order a(0 5 a < 1) in U, and the l—spirallike functions of order a(0 5 a < 1) in U. We refer to the monograph of Srivastava and Owa [5] (see also [2]) for the definitions (and other related details) of the above subclasses. In this paper, we consider a certain class of rational functions defined by ! 1 X z k F ðzÞ ¼ k = 0; jbk j 6 1; when k > 0 ; ð1:2Þ k P1 1 þ k¼1 bk zk k¼1
*
Corresponding author. E-mail addresses: [email protected] (R.K. Raina), [email protected] (D.K. Bansal).
0096-3003/$ - see front matter Ó 2006 Published by Elsevier Inc. doi:10.1016/j.amc.2006.08.139
404
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
which is analytic in the unit disk U. It is observed that when k ¼ 1;
bn 6¼ 0;
bnþj ¼ 0
ðj 2 NÞ;
ð1:3Þ
ðbn 6¼ 0Þ:
ð1:4Þ
in (1.2), then GðzÞ ¼
z 1 þ b1 z þ þ bn z n
The class of functions G(z) was studied by Reade et al. [4]. On the other hand, when b1 ¼ b2 ¼ ¼ bn1 ¼ 0;
bn ¼ r;
bnþj ¼ 0
ðj 2 NÞ;
ð1:5Þ
in (1.2), then F k ðzÞ ¼ H k ðz; rÞ ¼
z k
ð1 þ rzn Þ
ðk = 0; j r j6 1Þ:
ð1:6Þ
For r = 1 in (1.6), the subclass denoted by fk(z) was treated as Koebe type function by Fukui et al. [1]. Further, when r ¼ 1; k ¼ rðr 2 NÞ in (1.6), the subclass denoted by fn,r(z) was investigated by Mitrinovic [3]. This paper presents the familiar geometric properties like the starlikeness, convexity and spirallikeness for the class of functions defined by (1.2). We also consider some relavent particular cases of our main results (Theorems 1–3 below) by mentioning few known (and new) results. 2. Main results The starlikeness property satisfied by the class of functions Fk(z) defined by (1.2) is contained in Theorem 1 below. Theorem 1. Let Fk(z) be defined by (1.2). Then F k ðzÞ 2 S ðaÞð0 5 a < 1Þ, provided that 1 X fkk þ jkk 2ð1 aÞjg jbk j 5 2ð1 aÞ:
ð2:1Þ
k¼1
Proof. Let the inequality (2.1) be satisfied for the function Fk(z) defined by (1.2). In order to prove that F k ðzÞ 2 S ðaÞð0 5 a < 1Þ, it is sufficient to show that 1 zfF k ðzÞg0 k fF ðzÞg < 1 ðz 2 UÞ: ð2:2Þ zfF k ðzÞg0 1 2a þ fF k ðzÞg Using (1.2), we infer that the above inequality (2.2) holds true under the constraints given by (2.1), which evidently proves our Theorem 1. Our next result gives the sufficiency conditions for the class of functions Fk(z) defined by (1.2) to belong to KðaÞð0 5 a < 1Þ. h Theorem 2. Let Fk(z) be defined by (1.2). Then F k ðzÞ 2 KðaÞð0 5 a < 1Þ, provided that there exist numbers p, q > 0 such that 1p þ 1q 5 1, satisfying the inequalities: 1 X fpkðk þ 1Þ þ 1 agjbk j 5 1 a ð2:3Þ k¼1
and 1 X k¼1
ðqk þ 1 aÞjkk 1jjbk j 5 1 a:
ð2:4Þ
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
405
Proof. Making use of (1.2), and employing elementary calculations, we obtain P P1 k k zfF k ðzÞg00 ðk þ 1Þ 1 k¼1 kbk z k¼1 kðkk 1Þbk z P P ¼1 þ 1þ : k k 1þ 1 1 1 fF k ðzÞg0 k¼1 bk z k¼1 ðkk 1Þbk z For proving that F k ðzÞ 2 KðaÞð0 5 a < 1Þ, it is sufficient to show that zfF k ðzÞg00 R 1þ = a ðz 2 UÞ: fF k ðzÞg0 It readily follows that P P1 k k zfF k ðzÞ00 g ðk þ 1Þ 1 k¼1 kbk z k¼1 kðkk 1Þbk z P P þ ¼1R R 1þ k k 1þ 1 1 1 fF k ðzÞg0 k¼1 bk z k¼1 ðkk 1Þbk z P1 P 1 k ðk þ 1Þ k¼1 kbk zk k¼1 kðkk 1Þbk z P1 P1 = 1 k 1 þ k¼1 bk z 1 k¼1 ðkk 1Þbk zk and, in view of (2.3) and (2.4), we also infer that P P1 k ðk þ 1Þ 1 ðk þ 1Þ k¼1 kjbk j 1a k¼1 kbk z P P 1 þ 1 b zk 5 1 1 jb j 5 p k¼1 k
and
k¼1
ð2:5Þ
ð2:6Þ
ð2:7Þ
k
P1 P1 k 1a k¼1 kðkk 1Þbk z k¼1 kjkk 1jjbk j 1 P1 ðkk 1Þb zk 5 1 P1 jkk 1jjb j 5 q : k k k¼1 k¼1
ð2:8Þ
The inequality (2.6), in conjunction with (2.7) and (2.8) and the condition for p, q stated in the hypothesis of Theorem 2 establishes (2.5), which proves Theorem 2. Lastly, we obtain the spirallikeness property satisfied by the class of functions Fk(z) defined by (1.2), and this result is contained in Theorem 3 below. h Theorem 3. Let Fk(z) be defined by (1.2). Then Fk(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), provided that 1 X fj1 eil þ kkeil j þ j1 2a cos l þ ð1 kkÞeilj gjbk j 5 j1 2a cos l þ ei lj j1 eil j:
ð2:9Þ
k¼1
Proof. Let the inequality (2.9) be satisfied for the function Fk(z) defined by (1.2). To prove that Fk(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), it is sufficient to show that k ðzÞg0 1 eil zfF fF k ðzÞg 0 < 1 k ðzÞg ð1 2a cos lÞ þ eil zfF k fF ðzÞg
ðz 2 U; jlj < p=2; 0 5 a < 1Þ:
ð2:10Þ
Simple calculations reveal that the above inequality (2.10) holds true under the constraints (2.9), which proves Theorem 3. h 3. Some consequences of main results In this concluding section, we consider some consequences of our main results (Theorems 1–3) proved in Section 2. On setting k = 1, Theorem 1 would yield the following result: Corollary 1. Let F1(z) be defined by (1.2), then F 1 ðzÞ 2 S ðaÞ ð0 5 a < 1Þ, provided that 1 X ð1 aÞ1 jb1 j ð0 5 a 5 1=2Þ ðk 1 þ aÞjbk j 5 ð1 aÞ ajb1 j ð1=2 < a < 1Þ: k¼2 Corollary 1 was earlier obtained by Reade et al. [4, Theorem 2, p. 101].
ð3:1Þ
406
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
Next, if we set the parameters of the function defined by (1.2) in accordance with (1.5), then view of (1.6), Theorem 1 gives the result: Corollary 2. Let Hk(z;r) be defined by (1.6), then H k ðz; rÞ 2 S ðaÞ provided that fnk þ jnk 2ð1 aÞjgjrj 5 2ð1 aÞ:
ð3:2Þ
When r = 1, a = 0, the inequality (3.2) reduces to 0 5 nk 5 2, and the corresponding result from Corollary 2 is contained in a result given by Fukui et al. [1, Theorem 1, p. 109]. Also, when r = 1, a = 0, and k ¼ rðr 2 NÞ, Corollary 2 gives Corollary 3. Let fn ; rðzÞ ¼ ð1þzz n Þr ðr 2 NÞ, then fn ; rðzÞ 2 S , provided that 0 5 nr 5 2:
ð3:3Þ
Putting k = 1, then Theorem 2 yields the following result: Corollary 4. Let F1(z) be defined by (1.2). Then F 1 ðzÞ 2 KðaÞ, provided that there exist numbers p, q > 0 such that 1p þ 1q 5 1, satisfying the inequalities: 1 X
f2pk þ 1 agjbk j 5 1 a
ð3:4Þ
fqk þ 1 agðk 1Þjbk j 5 1 a:
ð3:5Þ
k¼1
and 1 X k¼1
Corollary 4 (when a = 0) was obtained by Reade et al. [4, Theorem 4, p. 102]. In Theorem 2, if k > 2 and q = p(k + 1), then the summation involved in (2.4) is observed to be greater than kþ2 the summation involved in (2.3). Hence, if we set p ¼ kþ1 ; q ¼ k þ 2, we are led to the following: Corollary 5. Let Fk(z)(k = 2) be defined by (1.2). Then F k ðzÞ 2 KðaÞ, provided that 1 X fkðk þ 2Þ þ 1 agðkk 1Þjbk j 5 1 a:
ð3:6Þ
k¼1
If we choose l = 0 in Theorem 3, then one gets immediately Theorem 1. Lastly, for k = 1, Theorem 3 gives Corollary 6. Let F1(z) be defined by (1.2). Then F1(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), provided that 1 X fj1 þ ðk 1Þeil j þ j1 2a cos l þ ð1 kÞeil jgjbk j 5 j1 2a cos l þ eil j j1 eil j:
ð3:7Þ
k¼1
Obviously, when l = 0, Corollary 6 would correspond to Corollary 1. Acknowledgement The present investigation was supported by AICTE (Govt. of India), New Delhi. References [1] S. Fukui, S. Owa, K. Sakaguchi, Some properties of analytic functions of Koebe type, in: H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992, p. 106. [2] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Deker, New York, 2000.
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
407
[3] D.S. Mitrinovic´, On the univalence of rational functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634–677 (1979) 221– 227. [4] M.O. Reade, H. Silverman, P.G. Todorov, Classes of rational functions, in: D.B. Shaffer (Ed.), Contemporary Mathematics: Topics in Complex Analysis, 38, American Mathematical Society, Providence, Rhode Island, 1985, pp. 99–103. [5] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.
Some properties of a certain class of rational functions R.K. Raina
a,*
, D.K. Bansal
b
a
b
Department of Mathematics, College of Technology and Agricultural Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur 313 001, Rajasthan, India Department of Mathematics, College of Science, Mohan Lal Sukhadia University, Udaipur 313 001, Rajasthan, India
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract In the present paper, we investigate the familiar geometric properties of starlikeness, convexity and spirallikeness of a certain class of rational functions. Relevance with various known (and new) results are also mentioned. Ó 2006 Published by Elsevier Inc. Keywords: Analytic functions; Starlike functions; Convex functions; Spirallike functions
1. Introduction Let A be the class of functions f(z) defined by 1 X ak z k ; f ðzÞ ¼ z þ
ð1:1Þ
k¼2
which are analytic in the open unit disk U ¼ fz : z 2 C; jzj < 1g. The subclasses of the class A denoted by S ðaÞ; KðaÞ and Sp(l, a) are, respectively, the well known subclasses of starlike functions of order a(0 5 a < 1) in U, the convex functions of order a(0 5 a < 1) in U, and the l—spirallike functions of order a(0 5 a < 1) in U. We refer to the monograph of Srivastava and Owa [5] (see also [2]) for the definitions (and other related details) of the above subclasses. In this paper, we consider a certain class of rational functions defined by ! 1 X z k F ðzÞ ¼ k = 0; jbk j 6 1; when k > 0 ; ð1:2Þ k P1 1 þ k¼1 bk zk k¼1
*
Corresponding author. E-mail addresses: [email protected] (R.K. Raina), [email protected] (D.K. Bansal).
0096-3003/$ - see front matter Ó 2006 Published by Elsevier Inc. doi:10.1016/j.amc.2006.08.139
404
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
which is analytic in the unit disk U. It is observed that when k ¼ 1;
bn 6¼ 0;
bnþj ¼ 0
ðj 2 NÞ;
ð1:3Þ
ðbn 6¼ 0Þ:
ð1:4Þ
in (1.2), then GðzÞ ¼
z 1 þ b1 z þ þ bn z n
The class of functions G(z) was studied by Reade et al. [4]. On the other hand, when b1 ¼ b2 ¼ ¼ bn1 ¼ 0;
bn ¼ r;
bnþj ¼ 0
ðj 2 NÞ;
ð1:5Þ
in (1.2), then F k ðzÞ ¼ H k ðz; rÞ ¼
z k
ð1 þ rzn Þ
ðk = 0; j r j6 1Þ:
ð1:6Þ
For r = 1 in (1.6), the subclass denoted by fk(z) was treated as Koebe type function by Fukui et al. [1]. Further, when r ¼ 1; k ¼ rðr 2 NÞ in (1.6), the subclass denoted by fn,r(z) was investigated by Mitrinovic [3]. This paper presents the familiar geometric properties like the starlikeness, convexity and spirallikeness for the class of functions defined by (1.2). We also consider some relavent particular cases of our main results (Theorems 1–3 below) by mentioning few known (and new) results. 2. Main results The starlikeness property satisfied by the class of functions Fk(z) defined by (1.2) is contained in Theorem 1 below. Theorem 1. Let Fk(z) be defined by (1.2). Then F k ðzÞ 2 S ðaÞð0 5 a < 1Þ, provided that 1 X fkk þ jkk 2ð1 aÞjg jbk j 5 2ð1 aÞ:
ð2:1Þ
k¼1
Proof. Let the inequality (2.1) be satisfied for the function Fk(z) defined by (1.2). In order to prove that F k ðzÞ 2 S ðaÞð0 5 a < 1Þ, it is sufficient to show that 1 zfF k ðzÞg0 k fF ðzÞg < 1 ðz 2 UÞ: ð2:2Þ zfF k ðzÞg0 1 2a þ fF k ðzÞg Using (1.2), we infer that the above inequality (2.2) holds true under the constraints given by (2.1), which evidently proves our Theorem 1. Our next result gives the sufficiency conditions for the class of functions Fk(z) defined by (1.2) to belong to KðaÞð0 5 a < 1Þ. h Theorem 2. Let Fk(z) be defined by (1.2). Then F k ðzÞ 2 KðaÞð0 5 a < 1Þ, provided that there exist numbers p, q > 0 such that 1p þ 1q 5 1, satisfying the inequalities: 1 X fpkðk þ 1Þ þ 1 agjbk j 5 1 a ð2:3Þ k¼1
and 1 X k¼1
ðqk þ 1 aÞjkk 1jjbk j 5 1 a:
ð2:4Þ
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
405
Proof. Making use of (1.2), and employing elementary calculations, we obtain P P1 k k zfF k ðzÞg00 ðk þ 1Þ 1 k¼1 kbk z k¼1 kðkk 1Þbk z P P ¼1 þ 1þ : k k 1þ 1 1 1 fF k ðzÞg0 k¼1 bk z k¼1 ðkk 1Þbk z For proving that F k ðzÞ 2 KðaÞð0 5 a < 1Þ, it is sufficient to show that zfF k ðzÞg00 R 1þ = a ðz 2 UÞ: fF k ðzÞg0 It readily follows that P P1 k k zfF k ðzÞ00 g ðk þ 1Þ 1 k¼1 kbk z k¼1 kðkk 1Þbk z P P þ ¼1R R 1þ k k 1þ 1 1 1 fF k ðzÞg0 k¼1 bk z k¼1 ðkk 1Þbk z P1 P 1 k ðk þ 1Þ k¼1 kbk zk k¼1 kðkk 1Þbk z P1 P1 = 1 k 1 þ k¼1 bk z 1 k¼1 ðkk 1Þbk zk and, in view of (2.3) and (2.4), we also infer that P P1 k ðk þ 1Þ 1 ðk þ 1Þ k¼1 kjbk j 1a k¼1 kbk z P P 1 þ 1 b zk 5 1 1 jb j 5 p k¼1 k
and
k¼1
ð2:5Þ
ð2:6Þ
ð2:7Þ
k
P1 P1 k 1a k¼1 kðkk 1Þbk z k¼1 kjkk 1jjbk j 1 P1 ðkk 1Þb zk 5 1 P1 jkk 1jjb j 5 q : k k k¼1 k¼1
ð2:8Þ
The inequality (2.6), in conjunction with (2.7) and (2.8) and the condition for p, q stated in the hypothesis of Theorem 2 establishes (2.5), which proves Theorem 2. Lastly, we obtain the spirallikeness property satisfied by the class of functions Fk(z) defined by (1.2), and this result is contained in Theorem 3 below. h Theorem 3. Let Fk(z) be defined by (1.2). Then Fk(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), provided that 1 X fj1 eil þ kkeil j þ j1 2a cos l þ ð1 kkÞeilj gjbk j 5 j1 2a cos l þ ei lj j1 eil j:
ð2:9Þ
k¼1
Proof. Let the inequality (2.9) be satisfied for the function Fk(z) defined by (1.2). To prove that Fk(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), it is sufficient to show that k ðzÞg0 1 eil zfF fF k ðzÞg 0 < 1 k ðzÞg ð1 2a cos lÞ þ eil zfF k fF ðzÞg
ðz 2 U; jlj < p=2; 0 5 a < 1Þ:
ð2:10Þ
Simple calculations reveal that the above inequality (2.10) holds true under the constraints (2.9), which proves Theorem 3. h 3. Some consequences of main results In this concluding section, we consider some consequences of our main results (Theorems 1–3) proved in Section 2. On setting k = 1, Theorem 1 would yield the following result: Corollary 1. Let F1(z) be defined by (1.2), then F 1 ðzÞ 2 S ðaÞ ð0 5 a < 1Þ, provided that 1 X ð1 aÞ1 jb1 j ð0 5 a 5 1=2Þ ðk 1 þ aÞjbk j 5 ð1 aÞ ajb1 j ð1=2 < a < 1Þ: k¼2 Corollary 1 was earlier obtained by Reade et al. [4, Theorem 2, p. 101].
ð3:1Þ
406
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
Next, if we set the parameters of the function defined by (1.2) in accordance with (1.5), then view of (1.6), Theorem 1 gives the result: Corollary 2. Let Hk(z;r) be defined by (1.6), then H k ðz; rÞ 2 S ðaÞ provided that fnk þ jnk 2ð1 aÞjgjrj 5 2ð1 aÞ:
ð3:2Þ
When r = 1, a = 0, the inequality (3.2) reduces to 0 5 nk 5 2, and the corresponding result from Corollary 2 is contained in a result given by Fukui et al. [1, Theorem 1, p. 109]. Also, when r = 1, a = 0, and k ¼ rðr 2 NÞ, Corollary 2 gives Corollary 3. Let fn ; rðzÞ ¼ ð1þzz n Þr ðr 2 NÞ, then fn ; rðzÞ 2 S , provided that 0 5 nr 5 2:
ð3:3Þ
Putting k = 1, then Theorem 2 yields the following result: Corollary 4. Let F1(z) be defined by (1.2). Then F 1 ðzÞ 2 KðaÞ, provided that there exist numbers p, q > 0 such that 1p þ 1q 5 1, satisfying the inequalities: 1 X
f2pk þ 1 agjbk j 5 1 a
ð3:4Þ
fqk þ 1 agðk 1Þjbk j 5 1 a:
ð3:5Þ
k¼1
and 1 X k¼1
Corollary 4 (when a = 0) was obtained by Reade et al. [4, Theorem 4, p. 102]. In Theorem 2, if k > 2 and q = p(k + 1), then the summation involved in (2.4) is observed to be greater than kþ2 the summation involved in (2.3). Hence, if we set p ¼ kþ1 ; q ¼ k þ 2, we are led to the following: Corollary 5. Let Fk(z)(k = 2) be defined by (1.2). Then F k ðzÞ 2 KðaÞ, provided that 1 X fkðk þ 2Þ þ 1 agðkk 1Þjbk j 5 1 a:
ð3:6Þ
k¼1
If we choose l = 0 in Theorem 3, then one gets immediately Theorem 1. Lastly, for k = 1, Theorem 3 gives Corollary 6. Let F1(z) be defined by (1.2). Then F1(z) 2 Sp(l, a) (0 5 a < 1,jlj < p/2), provided that 1 X fj1 þ ðk 1Þeil j þ j1 2a cos l þ ð1 kÞeil jgjbk j 5 j1 2a cos l þ eil j j1 eil j:
ð3:7Þ
k¼1
Obviously, when l = 0, Corollary 6 would correspond to Corollary 1. Acknowledgement The present investigation was supported by AICTE (Govt. of India), New Delhi. References [1] S. Fukui, S. Owa, K. Sakaguchi, Some properties of analytic functions of Koebe type, in: H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992, p. 106. [2] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Deker, New York, 2000.
R.K. Raina, D.K. Bansal / Applied Mathematics and Computation 187 (2007) 403–407
407
[3] D.S. Mitrinovic´, On the univalence of rational functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634–677 (1979) 221– 227. [4] M.O. Reade, H. Silverman, P.G. Todorov, Classes of rational functions, in: D.B. Shaffer (Ed.), Contemporary Mathematics: Topics in Complex Analysis, 38, American Mathematical Society, Providence, Rhode Island, 1985, pp. 99–103. [5] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.