On New Subclasses of Analytic Functions with Respect to Conjugate and Symmetric Conjugate Points

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1 Global Journal of Pure Applied Mathematics. ISSN Volume, Number 3 06, pp Research India Publications On New Subclasses of Analytic Functions with Respect to Conjugate Symmetric Conjugate Points Ajab Bai Akbarally Nurul Atikah Mohd Isa Department of Mathematics, Faculty of Computers Mathematical Sciences, Universiti Teknologi MARA, Shah Alam Selangor Malaysia. Abstract Let G c α, β G sc α, β be the generalized classes of analytic functions with respect to conjugate symmetric conjugate points of order δ where functions in both classes satisfy the conditions z E zf z αz f z Re α f z f z αz >δ, f z f z zf z αz f z Re α f z f z αz >δ, f z f z z E respectively for 0 δ< 0 α<. In this paper, the coefficient estimates the upper bounds for second Hankel determinant for both classes are determined. AMS subject classification: Primary 30C45. Keywords: Analytic functions, conjugate points, symmetric conjugate points, coefficient estimates, second Hankel determinant.

2 850 Ajab Bai Akbarally Nurul Atikah Mohd Isa. Introduction Let S be the class of normalized, analytic univalent functions in the unit disk E, z < written as fz z a n z n.. n Set P is the set of all functions that can be represented in the form pz a n z n.. that are regular in E, such that for z in E, Refz>0. Let Ss be the subclass of S consisting of the functions given by. such that zf z Re > 0,z E. f z f z This class was defined by Sakaguchi in [], called the class of starlike functions with respect to symmetric points. El-Ashwah Thomas [], introduced functions f S to be starlike with respect to conjugate points denoted by f Sc starlike with respect to symmetric conjugate points denoted by f Sc if zf z Re > 0, z E, f z f z zf z Re > 0, z E f z f z respectively. Halim [3] then defined the class of functions starlike with respect to conjugate points of order δ denoted by Sc δ the class of functions starlike with respect to symmetric conjugate points of order δ denoted by Ssc δ, where functions for both classes satisfy zf z Re >δ, z E, f z f z n zf z Re >δ, f z f z for 0 δ<respectively. z E

3 On New Subclasses of Analytic Functions In terms of subordination, Goel Mehrok in [4] generalized the Sakaguchi class by introducing the class Ss A, B as the class of starlike functions with respect to symmetric points, such that the functions in this class satisfy zf z f z f z Az Bz, z E, where B<A ; A, B, are the arbitrary fixed numbers. Subsequently, Dahhar Janteng [5], generalized both El-Ashwah Thomas [] Goel Mehrok [4] class by introducing Sc A, B, the class of starlike functions with respect to conjugate points. The class Ssc A, B is the class of starlike functions with respect to symmetric conjugate points which was defined by Ping Janteng [6]. We define G c α, δ as the class of analytic functions with respect to conjugate points of order δ G sc α, δ as the class of analytic functions with respect to symmetric conjugate points of order δ. The function f S given by. in this class must satisfy the conditions zf z αz f z Re α f z f z αz >δ, z E.3 f z f z zf z αz f z Re α f z f z αz >δ,z E.4 f z f z respectively for 0 δ< 0 α<. These classes are inspired by Selvaraj Vasanthi [7].. Coefficient Estimates We shall require the following lemma in order to prove our results. Lemma.. Pommerenke [8] If P p z is given by., then p n for n,, 3,... We give the coefficient inequalities for the class G c α, δ G sc α, δ. Theorem.. Let f G c α, δ, then for n, 0 δ< 0 α<, a n n n δ n α j δ,.5 j δ a n n j δ n δ nα j δ.6

4 85 Ajab Bai Akbarally Nurul Atikah Mohd Isa Proof. From.3., we have z α z 3a 3 z 3 α a z 6a 3 z 3 α z α z a 3 z 3 α z 4a z 6a 3 z 3 δ p z p z Equating the coefficients of like powers of z we have p a δ α, p δ p a 3 δ α δ, a 4 p 3 δ p p δ p p δ p, 3 δ 3α δ δ a 5 p 4 δ δ p p 3 δ p δ p p 4 δ 4α δ δ p p 3 δ p p δ p p δ p. 4 δ 4α3 δ δ δ Thus, we can conclude,.7 n δ n α a n p n α a p n n α a n p.8 n δ nα a n p n α a p n α a n p.9 Utilizing Lemma., we have p p p 3 p 4, thus, from.7, a δ α, a 3 a 4 4 δ 3α δ δ, a 5 3 δ δ α δ, 5 δ 4α δ δ..0 It follows that.5.6 hold for n,. We now prove.5 using induction. Equation.8 in conjunction with Lemma. yield a n n δ n α [ ] n n k α a k kα a k.. k k

5 On New Subclasses of Analytic Functions We assume that.5 holds for k 3, 4,...,n. Then from., we obtain n k a n k j δ k δ j δ n δ n α n k k k δ In order to complete the proof, it is sufficient to show that m δ m α m k k j δ k δ j δ m m δ m α n k j δ.. j δ k k δ j δ j δ j δ..3 j δ.3 is valid for m 3. Let us suppose that.3 is true for all m, 3<m n. Then from. n δ n α n k δ k n δ n δ n k k j δ j δ n k k k δ n δ n α k j δ k δ j δ n k k k δ j δ j δ j δ j δ n 4 n δ n α j δ n δ j δ n 3 n δ n α j δ n δ j δ

6 854 Ajab Bai Akbarally Nurul Atikah Mohd Isa n 4 n δ j δ n δ n α n δ j δ n 4 n δ n α j δ n δ j δ n 3 n δ n α j δ n δ j δ n δ n α n δ n 4 j δ n δ j δ n 3 n δ n α j δ n δ j δ n n δ n α j δ. j δ Thus.3 holds for m n hence.5 follows. Similarly we can prove.6. Remark.3. From Theorem., if we let α 0 δ 0, we obtain the results of Dahhar Janteng [5], by setting A B in their results. Then, if we set δ 0, we obtain the results of Selvaraj Vasanthi [7] by setting A B in their results. Theorem.4. Let f G sc α, δ, then for n, 0 δ< 0 α<, a n n j δ n n α j δ.4 a n n j δ n δ nα j δ.5 Proof. From.4. we have z α z 3a 3 z αz a 6a 3 z α z a 3 z 3 αz 6a 3 z 0a 5 z 4 δ p zp z

7 On New Subclasses of Analytic Functions Equating the coefficients of like powers of z, wehave a Thus, we can conclude, p α,a 3 a 4 p 3 δ p p 4 3α δ,a 5 p δ α, p 4 δ p 4 δ 4α δ..6 n n α a n p n α a 3 p n 3 n α a n p.7 n δ nα a n p n α a 3 p n n α a n p..8 Utilizing Lemma. we have p p p 3 p 4, thus from.6 we have that a a 5 α, a 3 δ α, a 4 4 δ 3α δ, 4α δ..9 It follows that.4.5 hold for n,. We prove.4 using induction. Equation.7 in conjunction with Lemma. yield [ ] an n kα a k..0 n n α We assume that.4 holds for k 3, 4,...,n. Then from.0, we obtain n a n.. n n α k k k k δ j δ j δ In order to complete the proof, it is sufficient to show that m k j δ m m α k δ j δ k m m α m j δ. j δ.

8 856 Ajab Bai Akbarally Nurul Atikah Mohd Isa. is valid for m 3. Let us suppose that. is true for all m, 3<m n. Then from. n k j δ n n α k δ j δ k n δ n n δ n α n k j δ k δ j δ k n n n α j δ n δ j δ n n δ j δ n n δ n α j δ n n n α j δ n δ j δ n j δ n δ n n α n δ j δ n n n α j δ. j δ Thus. holds for m n hence.4 follows. Similarly, we can prove Second Hankel Determinant The qth Hankel determinant for q n was first introduced by Noonan Thomas [9], a n a n a nq a n a n a nq H q n.... a nq a nq a nq The bounds have been investigated by many researchers for different classes. Such as in 983, Noor [0] studied the Hankel determinant problem for the class of functions with bounded boundary rotation Ehrenborg [], considered the Hankel determinant of exponential polynomials.

9 On New Subclasses of Analytic Functions Second Hankel determinant is obtained when setting q n, where H a a 3 a 3 a 4. The finding of the upper bounds of a a 4 a3 for the Hankel determinant started in the year 000. Janteng et al. [] studied the second Hankel determinant for the classes of starlike functions convex functions. Second Hankel determinant for starlike convex functions with respect to symmetric points were studied by Janteng et al. in [3]. In order to seek the upper bounds for a a 4 a3 for the class G c α, δ G sc α, δ, we shall require the following Lemmas. Lemma 3.. Pommerenke [8] If p P, then p k for k,, 3,... Lemma 3.. Libera Zlotkiewicz [4,5] Let the function p z P be given by.. Assume without restriction that p 0. By rewriting Lemma 3. for the cases n n 3, D p p p p p p 8 Re p p p 4p 0 which is equivalent to p p x 4 p for some x, x. Then D3 0is equivalent to 4p 3 4p p p 3 4 p p p p 4 p p p provides the relation 4p 3 p 3 4 p p x p 4 p x 4 p x z for some value of z, z. We give the upper bounds for the second Hankel determinant for the class G c α, δg sc α, δ. Theorem 3.3. If f z G c α, δ, then a a 4 a3 Proof. Since f z G c α, δ, so from.7 we have 4 α δ 3.3 p a δ α, p δ p a 3 δ α δ, p 3 a 4 p p 3 δ 3 δ 3α δ δ p 3 3 δ 3α. 3.4

10 858 Ajab Bai Akbarally Nurul Atikah Mohd Isa Then, a a 4 p 4 p p 3 δ 3 δ 3α α δ δ p p 3 3 δ 3α δ α Therefore, a3 p4 p p δ p δ α δ δ. a a 4 a3 p 4 p p 3 δ 3 δ 3α α δ δ p p 3 3 δ 3α δ α p4 p p δ p δ α δ δ [ ] where C α α δ 3α α3 δ p 4 [ ] 3 δ α δ 3α α3 δ δ p p [ ] [ α δ δ p p 3 3α α3 δ δ ] p C α 3α α α3 δ δ δ. 3.5 Then, [ α δ 3α α3 δ ] p 4 a a 4 a3 [ 3 δ α δ C α 3α α3 δ δ] p p [ α δ δ ] p p 3 [ 3α α3 δ δ ] p 3.6 Using Lemma 3. Lemma 3.3 in 3.6 we obtain [ α δ 3α α3 δ ] p 4 [3 δ α δ p 3α α3 δ δ]p 4 p x a a 4 a3 C α [ α δ δ] p 3 p 4 p p x p 4 p x 4 p x z 4 [ 3α α3 δ δ ] p 4 p x

11 On New Subclasses of Analytic Functions [ α δ δ δ 3α α3 δ 3] p 4 [ δ α 5 5δ δ 3α α3 δ δ]p 4 p x 4C α [p α δ. δ 3α α3 δ δ 4 p ] 4 p x [ α δ δ ] p 4 p x z 3.7 We now assume p p p [0, ]. Therefore, from 3.7 by using triangular inequality z, we have a a 4 a3 [ α δ δ δ 3α α3 δ 3] p 4 [ δ α 5 5δ δ 3α α3 δ δ ] p 4 p β 4C α [ p α δ δ 3α α3 δ δ 4 p ] 4 p β [ α δ δ ] p 4 p β Thus, a a 4 a3 4C α F β where β x [ α δ δ δ 3α α3 δ 3] p 4 [ δ α δ ] p 4 p [ δ α 5 5δ δ F β 3α α3 δ δ ] p 4 p β [ p α δ δp 3α α3 δ δ 4 p ] 4 p β Differentiating F β with respect to β, wehave F β [ δ α 5 5δ δ 3α α3 δ δ ] p 4 p [ p α δ δp 3α α3 δ δ 4 p ] 4 p β. Since F β > 0, then F β is an increasing. Hence, Max.F β F. Consequently, a a 4 a3 G p 3.8 4C α

12 860 Ajab Bai Akbarally Nurul Atikah Mohd Isa Now we let G p F. [ α δ δ δ 3α α3 δ 3] p 4 [ δ α δ ] p 4 p [ δ α 5 5δ δ G p 3α α3 δ δ ] p 4 p [ p α δ δp 3α α3 δ δ 4 p ] 4 p. After simplifying, we have G p [ α δ4 δ 3α α3 δ 5 3δ 3δ ] p 4 Then, where [ 4 δ α 3 δ3 δ 6 3α α3 δ δ ] p 6 3α α3 δ δ. G p A α p 4 B α p 6 3α α3 δ δ A α α δ4 δ 3α α3 δ 5 3δ 3δ B α 4 δ α 3 δ3 δ 6 3α α3 δ δ. By differentiating G p with respect to p, wehave G p 4A α p 3 B α p G p A α p B α. Letting G p 0, we obtain p 0 4 δ α 3 δ3 δ 6 3α α3 δ δ p α δ4 δ 3α α3 δ 5 3δ 3δ. Clearly, it shows that G p attains its maximum value at p 0. So, Max. G p G 0. Therefore, by substituting 3.5 G p such that p 0 into 3.8, we obtain a a 4 a3 4 α δ. Hence, by substituting p 0 into Lemma 3.3, we have p 0, p p 3 0.

13 On New Subclasses of Analytic Functions The results for G c α, δ is reduced to the result of Singh [6] when α 0 δ 0. Theorem 3.4. If f z G sc α, δ, then a a 4 a3 Proof. Since f z G sc α, δ, so from.6 we have 4 α δ 3.9 p a α, p a 3 δ α, p p a 4 4 3α δ p 3 4 3α Then, a a 4 p p 8 3α α δ p p 3 8 α 3α a 3 p α δ. Therefore, a a 4 a3 p p 8 3α α δ p p 3 8 α 3α p α δ δ α p C α p δ α p p 3 8 α 3α p where C α 8 3α α α δ. 3.3 Then, a a 4 a3 C α δ α p p δ α p p 3 8 α 3α p 3.3

14 86 Ajab Bai Akbarally Nurul Atikah Mohd Isa Using Lemma 3. Lemma 3.3 in 3.3 we obtain a a 4 a3 δ α p p 4 p x C α δ α p 3 4 p p x p 4 p x 4 p x z p 4 8 α 3α p 4 p x [ α δ4 δ 8 3α α ] p 4 [ δ α 3 δ 6 3α α ] p 4 p x 4C α [ p α δ 8 3α α 4 p ] 4 p x [ α δ ] p 4 p x. z 3.33 We now assume p p p [0, ]. Therefore, from 3.33 by using triangular inequality z, we have a a 4 a3 [ ] α δ4 δ 8 3α α p 4 [ ] δ α 3 δ 6 3α α p 4 p β [ 4C α p α δ 8 3α α 4 p ] 4 p β [ α δ ] p 4 p β [ ] α δ4 δ 8 3α α p 4 8p [ δ α p 3 δ α ] 4C α δ α 3 δ 6 3α α p 4 p β [ α δ p p 8 3α α 4 p ] 4 p β Thus, where a a 4 a3 β x 4C α F β F β [ α δ4 δ 8 3α α ] p 4 8p δ α p 3 δ α [ δ α 3 δ 6 3α α ] p 4 p β [ α δ p p 8 3α α 4 p ]

15 On New Subclasses of Analytic Functions Differentiating F β with respect to β, wehave F β [ δ α 3 δ 6 3α α ] p 4 p [ α δ p p 8 3α α 4 p ] 4 p β. Since F β > 0,then F β is an increasing. Hence, Max. F β F. Consequently, a a 4 a3 G p 4C α 3.34 Now we let G p F. { [ α G p δ ] p 4 [ 64 α 3α 4 α δ ] p 8 α 3α }. Then, where G p A α p 4 B α p 8 α 3α A α δ α B α 64 3α α 4 δ α. By differentiating G p with respect to p, wehave Letting G p 0 we obtain p 0 p G p 4A α p 3 B α p, G p A α p B α. 6 3α α δ α δ α. Clearly, it shows that G p attains its maximum value at p 0. So, Max. G p G 0. Therefore, by substituting 3.3 G p such that p 0 into 3.34, we obtain a a 4 a3 4 α δ. Hence, by substituting p 0 into Lemma 3.3, we have p 0, p p 3 0. When α 0, δ 0 α, δ 0 the results for G sc α, δ is reduced to the results of Singh [6].

16 864 Ajab Bai Akbarally Nurul Atikah Mohd Isa Acknowledgement The first author would like to thank Universiti Teknologi MARA the support of the grant 600-RMI/FRGS5/3/7/03 on the preparation of this paper. References [] Sakaguchi, K On a certain univalent mapping. Journal of the Mathematical Society of Japan,, [] El-Ashwah, R. M., & Thomas, D. K Some subclasses of close-to-convex functions. J. Ramanujan Math. Soc,, [3] Halim, S. A. 99. Functions starlike with respect to other points. International Journal of Mathematics Mathematical Sciences, 43, [4] Goel, R. M., & Mehrok, B. C. 98. A subclass of starlike functions with respect to symmetric points. Tamkang J. Math, 3, 4. [5] Dahhar, S. A. F. M., & Janteng, A A subclass of starlike functions with respect to conjugate points. Int. Math. Forum, 84, [6] Ping, L. C., & Janteng, A. 0. Subclass of starlike functions with respect to symmetric conjugate points. International Journal of Algebra, 56, [7] Selvaraj, C., & Vasanthi, N. 0. Subclasses of analytic functions with respect to symmetric conjugate points. Tamkang Journal of Mathematics, 4, [8] Pommerenke, C Univalent functions. Gottingen: Venhoeck & Ruprecht. [9] Noonan, J. W., & Thomas, D. K On the second Hankel determinant of a really mean p-valent functions. Transactions of the American Mathematical Society, 3, [0] Noor, K. I Hankel determinant problem for the class of functions with bounded boundary rotation. Revue Roumaine de Mathématiques Pures et Appliquées, 88, [] Ehrenborg, R The Hankel determinant of exponential polynomials. American Mathematical Monthly, [] Janteng, A., Halim, S. A., & Darus, M Hankel determinant for starlike convex functions. Int. J. Math. Anal., 3, [3] Janteng, A., Halim, S. A., & Darus, M Hankel determinant for functions starlike convex with respect to symmetric points. Journal of Quality Measurement Analysis,, [4] Libera, R. J. & Zlotkiewicz, E. J. 98. Early coefficients of the inverse of a regular convex function. Proceedings of the American Mathematical Society, 85, 5 30.

17 On New Subclasses of Analytic Functions [5] Libera, R. J., & Zlotkiewicz, E. J Coefficient bounds for the inverse of a function with derivative in positive real part. Proceedings of the American Mathematical Society, 87, [6] Singh, G. 03. Hankel determinant for analytic functions with respect to other points. Engineering Mathematics Letters,, 5 3.

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