Mathematica Moravica Vol. 22, No. 2 (2018), 83–88
doi: 10.5937/MatMor1802083C
Starlike functions of complex order with bounded radius rotation by using quantum calculus Asena Çetı˙ nkaya, Oya Mert Abstract. In the present paper, we study on the subclass of starlike functions of complex order with bounded radius rotation using q− difference operator denoted by Rk (q, b) where k ≥ 2, q ∈ (0, 1) and b ∈ C\{0}. We investigate coefficient inequality, distortion theorem and radius of starlikeness for the class Rk (q, b).
1. Introduction Let A be the class of functions f of the form (1)
f (z) = z +
∞ X
an z n ,
n=2
which are analytic in the open unit disc D := {z : |z| < 1} and satisfy the condition f (0) = f 0 (0)−1 = 0 for every z ∈ D. We say that f1 is subordinate to f2 , written as f1 ≺ f2 , if there exists a Schwarz function φ which is analytic in D with φ(0) = 0 and |φ(z)| < 1 such that f1 (z) = f2 (φ(z)). In particular, when f2 is univalent, then the subordination is equivalent to f1 (0) = f2 (0) and f1 (D) ⊂ f2 (D) (Subordination principle [4]). In 1909 and 1910, Jackson [5, 6] initiated a study of q− difference operator by f (z) − f (qz) Dq f (z) = , for z ∈ B {0}, (1 − q)z where B is a subset of complex plane C, called q− geometric set if qz ∈ B, whenever z ∈ B. Note that if a subset B of C is q− geometric, then it contains all geometric sequences {zq n }∞ 0 , zq ∈ B. Obviously, Dq f (z) → f 0 (z) as q → 1− . Note that such an operator plays an important role in the theory of hypergeometric series and quantum physics (see for instance [1, 3, 7]). 2010 Mathematics Subject Classification. 30C45. Key words and phrases. q-starlike function of complex order, bounded radius rotation, coefficient inequality, distortion theorem, radius of starlikeness. Full paper. Received 14 September 2018, revised 4 December 2018, accepted 8 December 2018, available online 15 December 2018. 83
c
2018 Mathematica Moravica
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Starlike Functions of complex order with bounded radius rotation. . .
For a function f (z) = z n , we observe that 1 − q n n−1 Dq z n = z . 1−q Therefore we have ∞ X 1 − q n n−1 Dq f (z) = 1 + an z , 1−q n=2
1−q n 1−q .
Clearly, as q → 1− , [n]q → n. where [n]q = Denote by Pq the family of functions of the form p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + . . . , analytic in D and satisfying the condition p(z) − 1 ≤ 1 , 1 − q 1 − q where q ∈ (0, 1) is a fixed real number. 1+z . This result is sharp Lemma 1.1 ([2]). p ∈ Pq if and only if p(z) ≺ 1−qz 1 + φ(z) for the functions p(z) = , where φ is a Schwarz function. 1 − qφ(z)
A function p analytic in D with p(0) = 1 is said to be in the class Pk (q), (1) (2) k ≥ 2, q ∈ (0, 1) if and only if there exists p1 , p2 ∈ Pq such that k 1 k 1 (1) (2) + p1 (z) − − p2 (z). p(z) = 4 2 4 2 For q → 1− , Pk (q) ≡ Pk , (see [10]); for k = 2, q → 1− , Pk (q) ≡ P is the well known class of functions with positive real part. Also, for k = 2, Pk (q) ≡ Pq 1+z consists of all functions subordinate to 1−qz , z ∈ D. Definition 1.1. Let f of the form (1) be an element of A. If f satisfies the condition Dq f (z) = p(z), p ∈ Pk (q), z f (z) with k ≥ 2, q ∈ (0, 1), then f is called q− starlike function with bounded radius rotation denoted by Rk (q). This class was introduced and studied by Noor et al. [9]. Definition 1.2. Let f of the form (1) be an element of A. If f satisfies the condition 1 Dq f (z) 1+ z − 1 = p(z), p ∈ Pk (q), b f (z) with k ≥ 2, q ∈ (0, 1), b ∈ C\{0}, then f is called q− starlike function of complex order with bounded radius rotation denoted by Rk (q, b). When q → 1− , b = 1, the class Rk (q, b) reduces to Rk (see [10]). For k = 2, q → 1− , the class Rk (q, b) reduces to S ∗ (1 − b) (see [8]). For k = 2, q → 1− , b = 1 the class Rk (q, b) reduces to traditional class of the starlike functions S ∗ .
Asena Çetı˙ nkaya, Oya Mert
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We investigate coefficient inequality, distortion theorem and radius of starlikeness for the class Rk (q, b). 2. Main Results We first prove coefficient inequality for the class Rk (q, b). For our main theorem, we need the following lemma. Lemma 2.1 ([11]). Let p(z) = 1 + p1 z + p2 z 2 + . . . be an element of Pk (q), then k |pn | ≤ (1 + q). 2 This result is sharp for the functions k 1 k 1 (1) p(z) = + p1 (z) − − p2 (z)(2) , 4 2 4 2 (1)
(2)
where p1 , p2 ∈ Pq . Theorem 2.1. If f ∈ Rk (q, b), then n−1 Y 1 k (2) |an | ≤ ([ν]q − 1) + |b|(1 + q) . ([n]q − 1)! 2 ν=1
This inequality is sharp for every n ≥ 2. Proof. In view of definition of the class Rk (q, b) and subordination principle, we can write 1 Dq f (z) z − 1 = p(z), 1+ b f (z) P n where p ∈ Pk (q) with p(0) = 1. Since f (z) = z + ∞ n=2 an z and p(z) = 2 1 + p1 z + p2 z + . . . , then we have z + [2]q a2 z 2 + [3]q a3 z 3 + [4]q a4 z 4 + . . . = z + (a2 + bp1 )z 2 + (a3 + bp1 a2 + bp2 )z 3 + (a4 + bp1 a3 + bp2 a2 + bp3 )z 4 + . . . . Comparing the coefficients of z n on both sides, we obtain [n]q an = an + bp1 an−1 + bp2 an−2 + · · · + bpn−2 a2 + bpn−1 for all integer n ≥ 2. In view of Lemma 2.1, we get ([n]q − 1)|an | ≤
k |b|(1 + q)(|an−1 | + · · · + |a2 | + 1), 2
or equivalently (3)
|an | ≤
k 2 |b|(1
n−1
+ q) X |aν |, [n]q − 1 ν=1
|a1 | = 1.
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Starlike Functions of complex order with bounded radius rotation. . .
In order to prove (2), we will use process of iteration. Let c = and use our assumption |a1 | = 1 in (3), we obtain successively for n = 2, for n = 3, for n = 4,
k 2 |b|(1
+ q)
1 c, [2]q − 1 1 |a3 | ≤ c(([2]q − 1) + c), ([3]q − 1)! 1 |a4 | ≤ c(([2]q − 1) + c)(([3]q − 1) + c). ([4]q − 1)!
|a2 | ≤
Hence induction shows that for n, we obtain |an | ≤
1 c(([2]q − 1) + c)(([3]q − 1) + c) · · · (([n − 1]q − 1) + c). ([n]q − 1)!
This proves (2). This inequality is sharp, because extremal function is the solution of the q− differential equation k 1 1+z k 1 1−z 1 Dq f (z) z −1 = + − − . 1+ b f (z) 4 2 1 − qz 4 2 1 + qz Corollary 2.1. Taking q → 1− and choosing k = 2, b = 1 in (2), we get |an | ≤ n for every n ≥ 2. This is well known coefficient inequality for starlike functions. We now introduce distortion theorem and radius of q− starlikeness for the class Rk (q, b). Lemma 2.2 ([9]). Let f ∈ Rk (q). Then for k ≥ 2 and q ∈ (0, 1), we have k Dq f (z) 1 + qr2 2 (1 + q)r z − ≤ . f (z) 2 2 1−q r 1 − q2 r2 Theorem 2.2. If f is an element of Rk (q, b), then 1−q−1 1−q−1 log q log q ≤ |f (z)| ≤ rF (k, q, Re b, |b|, r) , (4) rF (k, q, Re b, |b|, −r) where 1+q ( k |b|− 1 Re b) 2 1 + qr q 4 F (k, q, Re b, |b|, r) = 1+q (( k |b|+ 1 Re b) . 2 1 − qr q 4 This bound is sharp. Proof. In view of Lemma 2.2 and subordination principle, we write k 2 1 + 1 z Dq f (z) − 1 − 1 + qr ≤ 2 (1 + q)r . b f (z) 1 − q2 r2 1 − q2 r2
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Asena Çetı˙ nkaya, Oya Mert
Therefore, after routine calculations, we get Dq f (z) 1 + (b(q + q 2 ) − q 2 )r2 ≤ (5) z f (z) − 1 − q2 r2
k 2 |b|(1
+ q)r . 1 − q2 r2
After calculations in (5), we obtain Dq f (z) 1 + k2 |b|(1 + q)r + ((q 2 + q) Re b − q 2 )r2 ≤ , f (z) (1 − qr)(1 + qr) (6) Dq f (z) 1 − k2 |b|(1 + q)r + ((q 2 + q) Re b − q 2 )r2 Re z ≥ . f (z) (1 − qr)(1 + qr) On the other hand, we have Dq f (z) ∂q (7) Re z = r log |f (z)|. f (z) ∂r Re z
Considering (6) and (7) together, respectively, we get ∂q 1 (1 + q)( 21 Re b + k4 |b|) (1 + q)( 12 Re b − k4 |b|) log |f (z)| ≤ + − , ∂r r (1 − qr) (1 + qr) ∂q 1 (1 + q)( 21 Re b − k4 |b|) (1 + q)( 12 Re b + k4 |b|) log |f (z)| ≥ + − . ∂r r (1 − qr) (1 + qr) Taking q− integral on both sides of the last inequalities, we get (4). This bound is sharp, because extremal function is the solution of the q− differential equation 1 Dq f (z) k 1 1+z k 1 1−z 1+ z −1 = + − − . b f (z) 4 2 1 − qz 4 2 1 + qz Corollary 2.2. Taking q → 1− and b = 1 in Theorem 2.2, we get the following well known result: ( k −1) ( k −1) z 1+z 2 z 1−z 2 ( k +1) ≤ |f (z)| ≤ ( k +1) . 1+z 2 1−z 2 Let f ∈ A, then the real number Dq f (z) ∗ > 0 for all z ∈ D r (f ) = sup r > 0, Re z f (z) is called the starlikeness of the class A. The second inequality of (6) gives the starlikeness of the class Rk (q, b) as below: p k|b|(1 + q) − k 2 |b|2 (1 + q)2 − 16((q 2 + q) Re b − q 2 ) ∗ r (f ) = . 4((q 2 + q) Re b − q 2 ) √
2
If q → 1− , b = 1, then this radius reduces to r∗ (f ) = k− 2k −4 . This is the radius of the class Rk which was obtained by Pinchuk (see [10]). For q → 1− , k = 2, we get the starlikennes of starlike functions of complex order
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Starlike Functions of complex order with bounded radius rotation. . .
√ |b|− |b|2 −2 Re b+1 as rf∗ = . If k = 2, b = 1 then the radius of the starlikeness 2 Re b−1 ∗ ∗ of the class Sq is r (f ) = 1q . References [1] G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev., 16 (4) (1974), 441–484. [2] A. Çetinkaya, O. Mert, A certain class of harmonic mappings related to functions of bounded boundary rotation, Proc. of 12th Int. Sym. on Geometric Function Theory and Applications, (2016), 67–76. [3] N. J. Fine, Basic Hypergeometric series and applications, Math. Surveys Monogr., 1988. [4] A. W. Goodman, Univalent functions volume I and II, Polygonal Pub. House, 1983. [5] F. H. Jackson, On q− functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46 (2) (1909), 253–281. [6] F. H. Jackson, On q− definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203. [7] V. Kac, P. Cheung, Quantum calculus, Springer, 2002. [8] M. A Nasr, M. K. Aouf, Starlike functions of complex order, J. Natur. Sci. Math., 25 (1) (1985), 1–12. [9] K. I. Noor, M. A. Noor, Linear combinations of generaized q− starlike functions, Appl. Math. Inf. Sci., 11 (3) (2017), 745–748. [10] B. Pinchuk, Functions of bounded boundary Rotation, Isr. J. Math., 10 (1) (1971), 6–16. [11] Y. Polatoglu, A. Çetinkaya, O. Mert, Some properties of the analytic functions with bounded radius rotation, Creative Mathematics and Informatics, 28 (1) (2019), 6 pages.
Asena Çet˙ınkaya Department of Mathematics and Computer Sciences Istanbul Kültür University Istanbul Turkey E-mail address:
[email protected]
Oya Mert Department of Basic Sciences Altınbaş University Istanbul Turkey E-mail address:
[email protected] 
