Research article

Regions of variability for a subclass of analytic functions

  • Received: 31 December 2019 Accepted: 26 March 2020 Published: 31 March 2020
  • MSC : 30C45, 30C55, 30C80

  • Let $A\in \mathbb{C}, $ $B\in \lbrack -1, 0)\ $and $\alpha \in \left(-\frac{\pi }{2}, \frac{\pi }{2% }\right) $. Then $C_{\alpha }\left[ A, B\right] $ denotes the class of analytic functions $f$ in the open unit disc with \lt i \gt f \lt /i \gt (0) = 0 = \lt i \gt f \lt /i \gt '(0)-1 such that $ \begin{equation*} e^{i\alpha }\left(1+\frac{zf^{\prime \prime }\left(z\right) }{f^{\prime }\left(z\right) }\right) = \cos \alpha p\left(z\right) +i\sin \alpha, \end{equation*} $ with $ \begin{equation*} p\left(z\right) = \frac{1+Aw\left(z\right) }{1+Bw\left(z\right) }, \end{equation*} $ where $w\left(0\right) = 0$ and $\left\vert w\left(z\right) \right\vert \lt 1.$ Region of variability problems provides accurate information about a class of univalent functions than classical growth distortion and rotation theorems. In this article we find the regions of variability $V_{\lambda }\left(z_{0}, A, B\right) $ for $\log f^{\prime }\left(z_{0}\right) \ $when $f$ ranges over the class $C_{\alpha }\left[ \lambda, A, B\right] $ defined as $ \begin{equation*}{{C}_{\alpha }}\left[ \lambda, A, B \right] = \left\{ f\in {{C}_{\alpha }}\left[ A, B \right]:f''\left(0 \right) = \left(A-B \right){{e}^{-i\alpha }}\cos \alpha \right\}\end{equation*} $ for any fixed $z_{0}\in E$ and $\lambda \in \overline{E}$. As a consequence, the regions of variability are also illustrated graphically for different sets of parameters.

    Citation: Mohsan Raza, Wasim Ul Haq, Jin-Lin Liu, Saddaf Noreen. Regions of variability for a subclass of analytic functions[J]. AIMS Mathematics, 2020, 5(4): 3365-3377. doi: 10.3934/math.2020217

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  • Let $A\in \mathbb{C}, $ $B\in \lbrack -1, 0)\ $and $\alpha \in \left(-\frac{\pi }{2}, \frac{\pi }{2% }\right) $. Then $C_{\alpha }\left[ A, B\right] $ denotes the class of analytic functions $f$ in the open unit disc with \lt i \gt f \lt /i \gt (0) = 0 = \lt i \gt f \lt /i \gt '(0)-1 such that $ \begin{equation*} e^{i\alpha }\left(1+\frac{zf^{\prime \prime }\left(z\right) }{f^{\prime }\left(z\right) }\right) = \cos \alpha p\left(z\right) +i\sin \alpha, \end{equation*} $ with $ \begin{equation*} p\left(z\right) = \frac{1+Aw\left(z\right) }{1+Bw\left(z\right) }, \end{equation*} $ where $w\left(0\right) = 0$ and $\left\vert w\left(z\right) \right\vert \lt 1.$ Region of variability problems provides accurate information about a class of univalent functions than classical growth distortion and rotation theorems. In this article we find the regions of variability $V_{\lambda }\left(z_{0}, A, B\right) $ for $\log f^{\prime }\left(z_{0}\right) \ $when $f$ ranges over the class $C_{\alpha }\left[ \lambda, A, B\right] $ defined as $ \begin{equation*}{{C}_{\alpha }}\left[ \lambda, A, B \right] = \left\{ f\in {{C}_{\alpha }}\left[ A, B \right]:f''\left(0 \right) = \left(A-B \right){{e}^{-i\alpha }}\cos \alpha \right\}\end{equation*} $ for any fixed $z_{0}\in E$ and $\lambda \in \overline{E}$. As a consequence, the regions of variability are also illustrated graphically for different sets of parameters.


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    [1] O. P. Ahuja, H. Silverman, A survey on spiral-like and related function classes, Math. Chronicle, 20 (1991), 39-66.
    [2] J. Becker, Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354 (1984), 74-94.
    [3] J. H. Choi, Y. C. Kim, S. Ponnusamy, et al. Norm estimates for the Alexander transforms of convex functions of order alpha, J. Math. Anal. Appl., 303 (2005), 661-668. doi: 10.1016/j.jmaa.2004.08.066
    [4] S. Dineen, The Schwarz lemma, Oxford Math. Monogr., Clarendon Press Oxford, 1989.
    [5] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, New York, Berlin, Heidelberg, Tokyo, Springer-Verlag, 1983.
    [6] A. W. Goodman, Univalent Functions, Vols. I and II, Mariner Publishing Co. Tampa, Florida, 1983.
    [7] W. U. Haq, Variability regions for Janowski convex functions, Complex Var. Elliptic Equ., 59 (2014), 355-361. doi: 10.1080/17476933.2012.725164
    [8] J. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math., 28 (1973), 297-326. doi: 10.4064/ap-28-3-297-326
    [9] Y. C. Kim, S. Ponnusamy, T. Sugawa, Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives, J. Math. Anal. Appl., 299 (2004), 433-447. doi: 10.1016/j.jmaa.2004.03.081
    [10] Y. C. Kim, T. Sugawa, Correspondence between spirallike functions and starlike functions, Math. Nach., 285 (2012), 322-331. doi: 10.1002/mana.201010020
    [11] S. Ponnusamy, A. Vasudevarao, Regions of variability for functions with positive real part, Ann. Polon. Math., 99 (2010), 225-245. doi: 10.4064/ap99-3-2
    [12] S. Ponnusamy, A. Vasudevarao, Region of variability of two subclasses of univalent functions, J. Math. Anal. Appl., 332 (2007) 1323-1334.
    [13] S. Ponnusamy, A. Vasudevarao, M. Vuorinen, Region of variability for certain classes of univalent functions satisfying differential inequalities, Complex var. Elliptic Equ., 54 (2009), 899-922. doi: 10.1080/17476930802657616
    [14] S. Ponnusamy, A. Vasudevarao, H. Yanagihara, Region of variability for close-to-convex functions, Complex Var. Elliptic Equ., 53 (2008), 709-716. doi: 10.1080/17476930801996346
    [15] S. Ponnusamy, A. Vasudevarao, H. Yanagihara, Region of variability of univalent functions f (z) for which zf'(z) is spirallike, Houston J. Math., 34 (2008), 1037-1048.
    [16] M. Raza, W. U. Haq, S. Noreen, Regions of Variability for Janowski Functions, Miskloc Math. Notes, 16 (2015), 1117-1127. doi: 10.18514/MMN.2015.1344
    [17] M. S. Robertson, Univalent functions f (z) for which zf'(z) is spirallike, Mich. Math. J., 16 (1969), 97-101.
    [18] A. Y. Sen, Y. Polatoglu, M. Aydogan, Distortion theorem and the radius of convexity for Janowski-Robertson functions, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 291-294.
    [19] L. Spacek, Contribution a la theorie des fonctions univalentes (in Czech), Časopis Pest. Mat., 62 (1933), 12-19. doi: 10.21136/CPMF.1933.121951
    [20] S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J., 28 (1999), 217-230. doi: 10.14492/hokmj/1351001086
    [21] H. Yanagihara, Regions of variability for convex function, Math. Nach., 279 (2006), 1723-1730. doi: 10.1002/mana.200310449
    [22] H. Yanagihara, Regions of variability for functions of bounded derivatives, Kodai Math. J., 28 (2005), 452-462. doi: 10.2996/kmj/1123767023
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