Skip to main content

Hankel determinant problem of a subclass of analytic functions

Abstract

In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al.

2010 Mathematics Subject Classification: 30C45; 30C10.

1 Introduction

Let A be the class of analytic function satisfying the condition f (0) = 0, f'(0) - 1 = 0 in the open unit disc E= { z : z < 1 } . By S, S * , C, and K we means the well-known subclasses of A which consist of univalent, starlike, convex, and close-to-convex functions, respectively.

Let V k λ ( σ ) ,k2,0σ<1,λ real ,λ< π 2 , denote the class of functions f1(z) analytic and locally univalent in , f1(0) = 0, f 1 ( 0 ) =1 and satisfying

0 2 π Re e i λ ( z f 1 ( z ) ) f 1 ( z ) - σ cos λ ( 1 - σ ) d θ k π cos λ , z = r e i θ .
(1.1)

This class was introduced and studied in details by Moulis [1]. For λ = 0, we obtain the class V k ( σ ) of analytic functions with bounded boundary rotations of order σ studied by Padmanabhan et al. [2] and when σ = 0 and λ = 0, we get the class V k discussed by Paatero [3], see also [48]. Also it can easily be shown that f 1 ( z ) V k λ ( σ ) if and only if there exists f 2 ( z ) V k such that

f 1 ( z ) = ( f 2 ( z ) ) ( 1 - σ ) e - i λ cos λ .
(1.2)

We now consider a class of analytic functions defined by Arif et al. [9] as follows:

Definition 1.1. Let f ( z ) A in E. Then f ( z ) B ̃ k ( λ , σ , β , γ ) , if for k ≥ 2, 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1, λ is real with λ< π 2 there exists a function f 1 ( z ) V k λ ( σ ) , 0 ≤ σ < 1, such that

| arg { z 1 γ f ( z ) f ( z ) ( f ( z ) f 1 ( z ) ) γ } | β π 2 , z E .

By giving specific values to the parameters k, σ, λ, β, and γ in B ̃ k ( λ , σ , β , γ ) , we obtain many important subclasses studied by various authors in earlier articles, see [1016].

Using (1.1) and (1.2), we have

z f ( z ) = z γ ( f ( z ) ) 1 - γ ( f 1 ( z ) ) γ p β ( z ) ,
(1.3)

where f 1 ( z ) V k λ ( σ ) and p(z) belongs to the class P of functions whose real part is positive.

Throughout in this article, we shall assume, unless otherwise stated, that k ≥ 2, 0 ≤ β ≤ 1, 0 < γ ≤ 1, λ is real with λ< π 2 , 0 ≤ σ < 1.

In [17], the q th Hankel determinant H q ( n ) , q ≥ 1, n ≥ 1, for a function f ( z ) A is stated by Noonan and Thomas as:

Definition 1.2. Let f ( z ) A. Then the q th Hankel determinant of f (z) is defined for q ≥ 1, n ≥ 1 by

H q ( n ) = a n a n + 1 a n + q - 1 a n + 1 a n + 2 a n + q - 2 a n + q - 1 a n + q - 2 a n + 2 q - 2 .
(1.4)

The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [[18], p. 329], Edrei [19] and in the study of power series with integral coefficients by Polya [[20], p. 323], Cantor [21], and many others.

In this article, we shall determine the rate of growth of the Hankel determinant H q ( n ) for f ( z ) B ̃ k ( λ , σ , β , γ ) with 0 < β < 2, as n → ∞. This determinant has been considered by several authors. That is, Noor [22] determined the rate of growth of H q ( n ) as n → ∞ for a function f(z) belongs to the class V k . Pommerenke in [23] studied the Hankel determinant for starlike functions. The Hankel determinant problem for other interesting classes of analytic functions were discussed by Noor [11, 12, 24].

Lemma 1.1. Let f ( z ) A. Let the q th Hankel determinant of f (z) for q ≥ 1, n ≥ 1 be defined by (1.4). Then, writting Δ j (n) = Δ j (n, z1, f(z)), we have

H q ( n ) = Δ 2 q - 2 ( n ) Δ 2 q - 3 ( n + 1 ) Δ q - 1 ( n + q - 1 ) Δ 2 q - 3 ( n + 1 ) Δ 2 q - 4 ( n + 2 ) Δ q - 2 ( n + q - 2 ) Δ q - 1 ( n + q - 1 ) Δ q - 2 ( n + q - 2 ) Δ q ( n + 2 q - 2 ) ,

where with Δ0(n) = a n , we define for j ≥ 1,

Δ j ( n , z 1 , f ( z ) ) = Δ j - 1 ( n , z 1 , f ( z ) ) - Δ j - 1 ( n + 1 , z 1 , f ( z ) ) .
(1.5)

Lemma 1.2. With z 1 = n n + 1 y and v ≥ 0 any integer,

Δ j ( n + v , z 1 , z f ( z ) ) = m = 0 j j m y m ( v - ( m - 1 ) n ) ( n + 1 ) m Δ j - m ( n + m + v , f ( z ) ) .

Lemmas 1.1 and 1.2 are due to Noonan and Thomas [17].

Lemma 1.3. Let h1(z) be starlike univalent function in . Then

  1. (i)

    there exists a z1 with |z1| = r such that for all z, |z| = r

z - z 1 h 1 ( z ) 2 r 2 1 - r 2 ,

   see [25]

(ii)

r ( 1 + r ) 2 h 1 ( z ) r ( 1 + r ) 2 ,

   see[26].

2 Hankel determinant problem

Theorem 2.1. Let f ( z ) B ̃ k ( λ , σ , β , γ ) with 0 < β < 2 and let the q th Hankel determinant H q ( n ) of f(z) be defined as in (1.4). Then

H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 2 , q = 1 n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 q - q 2 , q 2 , k 8 ( q - 1 ) ( 1 - σ ) γ cos 2 λ - 2 ,

where k> 4 j + 2 - β ( 1 - σ ) γ cos 2 λ -2 and O(1) is a constant depending on k, λ, β, σ, γ, and j only.

Proof. It is well known [1] that for starlike functions h1(z) and h2(z)

f 1 ( z ) = ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ .
(2.1)

Using (2.1) in (1.3), we have

z f ( z ) = z γ ( f ( z ) ) 1 - γ ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p β ( z ) .
(2.2)

where p ( z ) P.

Let F (z) = zf'(z). Then for j ≥ 1, z1 any non-zero complex and z = re, consider Δ j (n, z1, F(z)) as defined by (1.5). Then

Δ j ( n , z 1 , F ( z ) ) = 1 2 π r n + j 0 2 π z - z 1 j F ( z ) e i ( n + j ) θ d θ ,

and by using (2.2), we have

Δ j ( n , z 1 , F ( z ) ) 1 2 π r n + j 0 2 π z - z 1 j z γ f ( z ) 1 - γ ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p ( z ) β d θ ( M ( r ) ) 1 - γ 2 π r n + j 0 2 π z - z 1 j ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p ( z ) β d θ .
(2.3)

Since

( h 1 ( z ) / z ) k 4 + 1 2 ( h 2 ( z ) / z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ ( h 1 ( z ) / z ) k 4 + 1 2 ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ e γ ( 1 - σ ) γ π 2 sin 2 λ 2 = ( h 1 ( z ) / z ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ c 1 , say

therefore (2.3) becomes

Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 π r n + j 0 2 π z - z 1 j ( h 1 ( z ) / z ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ p ( z ) β d θ .

Now using Lemma 1.3, we have

Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 π r n + j ( 1 + r ) 2 r k 4 - 1 2 γ ( 1 - σ ) cos 2 λ 1 r γ ( 1 - σ ) cos 2 λ 2 r 2 1 - r 2 j 0 2 π ( h 1 ( z ) ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ - j p ( z ) β d θ .

The well-known Holder's inequality will give us

Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j 1 2 π 0 2 π p ( z ) 2 d θ β 2 1 2 π 0 2 π | h 1 ( z ) | k 2 + 1 γ ( 1 - σ ) cos 2 λ - 2 j 2 - β d θ 2 - β 2 .
(2.4)

Also, it is known [15] that, for p ( z ) P, z E,

1 2 π 0 2 π p ( z ) 2 d θ 1 + 3 r 2 1 - r 2 .
(2.5)

Using (2.5) in (2.4), we obtain

Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j 1 + 3 r 2 1 - r 2 β 2 1 2 π 0 2 π | ( h 1 ( z ) ) | k 4 + 1 γ ( 1 - σ ) cos 2 λ - 2 j 2 - β d θ 2 - β 2 .

Therefore, we can write

Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + β + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j + β 2 r k 4 + 1 2 γ ( 1 - σ ) cos 2 λ - j 1 2 π 0 2 π d θ 1 - r e i θ ( k + 2 ) γ ( 1 - σ ) cos 2 λ - 4 j 2 - β 2 - β 2 .

Now using a subordination result for starlike functions, we have

Δ j ( n , z 1 , F ( z ) ) c 2 ( M ( r ) ) 1 - γ r n 1 1 - r j + β 2 1 1 - r γ ( k + 2 ) ( 1 - σ ) cos 2 λ - 4 j 2 - β - 1 2 - β 2 , = c 2 ( M ( r ) ) 1 - γ r n 1 1 - r γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 - j ,

where c2 is a constant depending on k, λ, β, σ, γ, j only and γ(k + 2) (1 - σ) cos2 λ - 4j > 2 - β.

Applying Lemma 1.2 and putting z 1 = n n + 1 e i θ n , ( n ) ,r=1- 1 n , we have for k 4 j - β + 2 ( 1 - σ ) γ cos 2 λ -2,

Δ j n , e i θ n , f ( z ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - j - 2 ,

where O(1) is a constant depending on k, λ, β, σ, γ, and j only.

We now estimate the rate of growth of H q ( n ) .

For q = 1, H q ( n ) = a n = Δ 0 ( n ) and

H 1 ( n ) = a n = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 2 .

For q ≥ 2, we use similar argument due to Noonan and Thomas [17] together with Lemma 1.1 to have

H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 q - q 2 , k 8 ( q - 1 ) ( 1 - σ ) γ cos 2 λ - 2

and O(1) depends only on k, λ, β, σ, γ, and j.

For choosing different values of λ, β, σ, and γ in Theorem 2.1, we obtain the following results discussed by Noor [10, 11] and Noor et al. [14].

Corollary 2.1. For λ = 0, β = 1, σ = 0, f ( z ) B k ( γ ) , where the class B k ( γ ) was introduced by Noor et al. [14] and

H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 - 1 , q = 1 n γ k 2 + 1 q - q 2 , q 2 , k 8 ( q - 1 ) γ - 2

where O(1) is a constant depending only on k and γ.

Corollary 2.2. For λ=0,γ=1,f ( z ) T ̃ k ( β , σ ) and

H q ( n ) = O ( 1 ) n ( 1 - σ ) k 2 + 1 + β - 2 , q = 1 n ( 1 - σ ) k 2 + 1 + β - 1 q - q 2 , q 2 , k 8 ( q - 1 ) ( 1 - σ ) - 2

where O(1) is a constant depends on k, σ, and β only.

References

  1. Moulis EJ: Generalizations of the Robertson functions. Pac J Math 1979, 81(1):167–174.

    MATH  MathSciNet  Article  Google Scholar 

  2. Padmanabhan K, Parvatham R: Properties of a class of functions with bounded boundary rotation. Ann Polon Math 1975, 31: 311–323.

    MathSciNet  Google Scholar 

  3. Paatero V: Uber Gebiete von beschrankter Randdrehung. Ann Acad Sci Fenn Ser A 1933, 37(9):20.

    Google Scholar 

  4. Noor MA, Noor KI, Al-Said EA: On certain analytic functions with bounded radius rotation. Comput Math Appl 2011, 61(10):2987–2993. 10.1016/j.camwa.2011.03.084

    MATH  MathSciNet  Article  Google Scholar 

  5. Noor KI, Haq W, Arif M, Mustafa S: On bounded boundary and bounded radius rotations. J Inequal Appl, vol 2009., 2009: Art. ID 813687, 12

    Google Scholar 

  6. Noor KI, Noor MA, Al-Said EA: On multivalent functions of bounded radius rotations. Appl Math Lett 2011, 24(7):1155–1159. 10.1016/j.aml.2011.01.042

    MATH  MathSciNet  Article  Google Scholar 

  7. Noor KI, Noor MA, Al-Said EA: On analytic functions of bounded boundary rotation of complex order. Comput Math Appl 2011, 62: 2112–2125. 10.1016/j.camwa.2011.06.059

    MATH  MathSciNet  Article  Google Scholar 

  8. Arif M, Noor KI, Raza M: On a class of analytic functions related with generalized Bazilevic type functions. Comput Math Appl 2011, 61: 2456–2462. 10.1016/j.camwa.2011.02.026

    MATH  MathSciNet  Article  Google Scholar 

  9. Arif M, Raza M, Noor KI, Malik SN: On strongly Bazilevic functions associated with generalized Robertson functions. Math Comput Model 2011, 54: 1608–1612. 10.1016/j.mcm.2011.04.033

    MATH  MathSciNet  Article  Google Scholar 

  10. Noor KI: On certain analytic functions related with strongly close-to-convex functions. Appl Math Comput 2008, 197: 149–157. 10.1016/j.amc.2007.07.039

    MATH  MathSciNet  Article  Google Scholar 

  11. Noor KI: On the Hankel determinant of close-to-convex univalent functions. Int J Math Math Sci 1980, 3: 447–481.

    Google Scholar 

  12. Noor KI: On the Hankel determinant problem for strongly close-to-convex functions. J Nat Geom 1997, 11: 29–34.

    MATH  Google Scholar 

  13. Noor KI: Some classes of analytic functions related with Bazilevic functions. Tamkang J Math 1997, 28(3):201–204.

    MATH  MathSciNet  Google Scholar 

  14. Noor KI, Al-Bany SA: On Bazilevic functions. Int J Math Math Sci 1987, 10(1):79–88. 10.1155/S0161171287000103

    MATH  MathSciNet  Article  Google Scholar 

  15. Pommerenke C: On close-to-convex analytic functions. Trans Am Math Soc 1965, 114: 176–186. 10.1090/S0002-9947-1965-0174720-4

    MATH  MathSciNet  Article  Google Scholar 

  16. Thomas DK: On Bazilevic functions. Trans Am Math Soc 1968, 132: 353–361.

    Google Scholar 

  17. Noonan JW, Thomas DK: On the Hankel determinant of areally mean p-valent functions. Proc Lond Math Soc 1972, 25(3):503–524. 10.1112/plms/s3-25.3.503

    MATH  MathSciNet  Article  Google Scholar 

  18. Dienes P: The Taylor series. Dover, New York; 1957.

    Google Scholar 

  19. Edrei A: Sur les determinants recurrents et less singularities d'une fonction donee por son developpement de Taylor. Compos Math 1940, 7: 20–88.

    MathSciNet  Google Scholar 

  20. Polya G, Schoenberg IJ: Remarks on de la Vallee Poussin means and convex conformal maps of the circle. Pac J Math 1958, 8: 259–334.

    MathSciNet  Article  Google Scholar 

  21. Cantor DG: Power series with integral coefficients. Bull Am Math Soc 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4

    MATH  MathSciNet  Article  Google Scholar 

  22. Noor KI: Hankel determinant problem for functions of bounded boundary rotations. Rev Roum Math Pures Appl 1983, 28: 731–739.

    MATH  Google Scholar 

  23. Pommerenke C: On starlike and close-to-convex analytic functions. Proc Lond Math Soc 1963, 13: 290–304. 10.1112/plms/s3-13.1.290

    MATH  MathSciNet  Article  Google Scholar 

  24. Noor KI, Al-Naggar IA: Hankel determinant problem. J Nat Geom 1998, 14(2):133–140.

    MATH  MathSciNet  Google Scholar 

  25. Golusin GM: On distortion theorem and coefficients of univalent functions. Math Sb 1946, 19: 183–203.

    MathSciNet  Google Scholar 

  26. Polatoglu Y, Bolcal M: Some results on the Janowski's starlike functions of complex order. arXiv:math/0007133v1

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Muhammad Arif.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MA, KIN, and MR jointly discussed and presented the ideas of this article. MA made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Arif, M., Noor, K.I. & Raza, M. Hankel determinant problem of a subclass of analytic functions. J Inequal Appl 2012, 22 (2012). https://doi.org/10.1186/1029-242X-2012-22

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-22

Keywords

  • Robertson function
  • strongly Bazilevic functions
  • bounded boundary rotations
  • Hankel determinant