Coefficient Inequalities of Second Hankel Determinants for Some Classes of Bi-Univalent Functions

mathematics Article Coefficient Inequalities of Second Hankel Determinants for Some Classes of Bi-Univalent Functions Rayaprolu Bharavi Sharma Kalikota Rajya Laxmi, * Department of Mathematics, Kakatiya University, Warangal, Telangana-506009, India; rbsharma005@gmail.com Department of Mathematics, SRIIT, Hyderabad, Telangana-5030, India * Correspondence: rajyalaxmi06@gmail.com; Tel.: +9-770-676-584. Academic Editor: Hari M. Srivastava Received: 7 December 05; Accepted: February 06; Published: 5 February 06 Abstract: In this paper, we investigate two sub-classes S pθ, βq K pθ, βq of bi-univalent functions in the open unit disc that are subordinate to certain analytic functions. For functions belonging to these classes, we obtain an upper bound for the second Hankel determinant H pq. Keywords: analytic functions; univalent functions; bi-univalent functions; second Hankel determinants. Introduction Let A be the class of the functions of the form f pzq z ` 8ÿ a k z k, () which are analytic in the open unit disc tz : z ă u. Further, by S we shall denote the class of all functions in A that are univalent in. Let P denote the family of functions p pzq, which are analytic in such that p p0q, Rp pzq ą 0 pz P q of the form 8ÿ P pzq ` c n z n. () For two functions f g, analytic in, we say that the function f is subordinate to g in, we write it as f pzq ă g pzq if there exists a Schwarz function ω, which is analytic in with ω p0q 0, ω pzq ă pz P q such that Indeed, it is known that k n f pzq g pω pzqq. (3) f pzq ă g pzq ñ f p0q g p0q f p q Ă g p q. (4) Every function f P S has an inverse f, which is defined by f p f pzqq z, pz P q f f pwq w, w ă r 0 p f q ; r 0 p f q ě. (5) 4 Mathematics 06, 4, 9; doi:0.3390/math400009 www.mdpi.com/journal/mathematics

Mathematics 06, 4, 9 of In fact, the inverse function is given by f pwq w a w ` a a 3 w 3 5a 3 5a a 3 ` a 4 w 4 `... A function f P A is said to be bi-univalent in if both f f are univalent in. Let ř denote the class of bi-univalent functions defined in the unit disc. We notice that ř is non empty.one of the best examples of bi-univalent functions is f pzq log ` z z, which maps the unit disc univalently onto a strip Imw ă π, which in turn z contains the unit disc. Other examples are z,, log p zq. z However, the Koebe function is not a member of ř because it maps unit disc univalent onto the entire complex plane minus a slit along to 8. Hence, the image domain does not contain the 4 unit disc. Other examples of univalent function that are not in the class ř are z z, z z. In 967, Lewin [] first introduced class ř of bi-univalent function showed that a ď.5 for every f P ř. Subsequently, in 967, Branan Clunie [] conjectured that a ď? for bi-star like functions a ď for bi-convex functions. Only the last estimate is sharp; equality occurs only for f pzq z or its rotation. z Later, Netanyahu [3] proved that max ř f P a 4. In 985, Kedzierawski [4] proved Brannan 3 Clunie s conjecture for bi-starlike functions. In 985, Tan [5] obtained that a ă.485, which is the best known estimate for bi-univalent functions. Since then, various subclasses of the bi-univalent function classes ř were introduced, non-sharp estimates on the first two coefficients a a 3 in the Taylor Maclaurin s series expansion were found in several investigations. The coefficient estimate problem for each of a n pn P N t, 3uq is still an open problem. In 976, Noonan Thomas [6] defined q th Hankel determinant of f for q ě n ě, which is stated by a n a n` a n`q a n` a n` a n`q H q pnq..... a n`q a n`q a n`q These determinants are useful, for example, in showing that a function of bounded characteristic in, i.e., a function that is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficient is rational. The Hankel determinant plays an important role in the study of singularities (for instance, see [7] Denies, p.39 Edrei [8]).A Hankel determinant plays an important role in the study of power series with integral coefficients. In 966, Pommerenke [9] investigated the Hankel determinants of areally mean p-valent functions, univalent functions as well as of starlike functions,, in 967 [0], he proved that the Hankel determinants of univalent functions satisfy H q pnq ă Kn p `βqq`3 pn,,..., q, 3,...q where β ą K depend only on q. 4000 Later, Hayman [] proved that H pnq ă An pn,,...; A an absolute constantq for areally mean univalent functions. The estimates for the Hankel determinant of areally mean p-valent functions have been investigated [ 4]. Elhosh [5,6] obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α k-fold symmetric close to convex functions. Noor [9] determined the rate of growth of H q pnq as n Ñ 8 for the functions in S with bounded boundary.

Mathematics 06, 4, 9 3 of Ehrenborg [7] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence some of its properties were discussed by Layman [8]. One can easily observe that the Fekete-Szego functional a3 a H pq. This function was further generalized with µ real as well as complex. Fekete-Szego gave a sharp estimate of a3 µa for µ real. The well-known results due to them is $ 4µ & 3 µ ď a 3 µa µ ď ` exp 0 ď µ ď µ % 3 4µ µ ě 0 On the other h, Zaprawa [9,0] extended the study on Fekete-Szego problem to some classes of bi-univalent functions. Ali [] found sharp bounds on the first four coefficients a sharp estimate for the Fekete-Szego functional γ 3 tγ, where t is real, for the inverse function of f defined as f pwq w ` 8ř γ k w k to the class of strongly starlike functions of order α p0 ă α ď q. k Recently S.K. Lee et al. [] obtained the second Hankel determinant H pq a a 4 a 3 for functions belonging to subclasses of Ma-Minda starlike convex functions. T. Ram Reddy [3] obtained the Hankel determinants for starlike convex functions with respect to symmetric points. T. Ram Reddy et al. [4,5] also obtained the second Hankel determinant for subclasses of p-valent functions p-valent starlike convex function of order α. Janteng [6] has obtained sharp estimates for the second Hankel determinant for functions whose derivative has a positive real part.afaf Abubaker [7] studied sharp upper bound of the second Hankel determinant of subclasses of analytic functions involving a generalized linear differential operator. In 05, the second Hankel determinant for bi-starlike bi-convex function of order β was obtained by Erhan Deniz [8].. Preliminaries Motivated by above work, in this paper, we introduce certain subclasses of bi-univalent functions obtained an upper bound to the coefficient functional a a 4 a 3 for the function f in these classes defined as follows: Definition..: A function f P A is said to be in the class S pθ, βq if it satisfies the following conditions: " " z f R e iθ ** pzq ą βcosθ p@z P q (6) f pzq " " wg R e iθ ** pwq ą βcosθ p@w P q (7) g pwq where g is an extension of f to. Note:. For θ 0, the class S pθ, βq reduces to the class S σ pβq,, for this class, coefficient inequalities of the second Hankel determinant were studied by Deniz et al [8].. For θ 0 β 0, the class S pθ, βq reduces to the class S σ,, for this class, coefficient inequalities of the second Hankel determinant were studied by Deniz et al [8]. Definition..: A function f P A is said to be in the class K pθ, βq if it satisfies the following conditions: " R "e iθ ` z f ** pzq f ą βcosθ p@z P q (8) pzq " ** R "e iθ ` wg pwq g ą βcosθ p@w P q (9) pwq where g is an extension of f in..

Mathematics 06, 4, 9 4 of Note:. For θ 0, the class K pθ, βq reduces to the class K σ pβq,, for this class, coefficient inequalities ofthe second Hankel determinantwere studied by Deniz et al [8].. For θ 0 β 0 the class K pθ, βq reduces to the class K σ,, for this class, coefficient inequalities ofthe second Hankel determinantwere studied by Deniz et al [8]. To prove our results, we require the following Lemmas: Lemma.. [4] Let the function p P P be given by the following series: p pzq ` c z ` c z ` c 3 z 3 `... pz P q. (0) Then the sharp estimate is given by. Lemma.. [9] The power series for the function p P P is given (0) converges in the unit disc to a function in P if only if Toeplitz determinants c c é c n c c é c n D n, n P N..... c n c n` c n` é c k c k are all non-negative. These are strictly positive except for p pzq ř m k ρ k P `eit 0 k z, ` z ρ k ą 0, t k real t k t j for k j, where P 0 pzq ; in this case, D n ą 0 for n ă pm q z D n 0 for n ě m. This necessary sufficient condition found in the literature [9] is due to Caratheodary Toeplitz. We may assume without any restriction that c ą 0. On using Lemma (.) for n n 3 respectively, we get It is equivalent to D c c c c c c c Then D 3 ě 0 is equivalent to! ) ı 8 ` Re c c c 4c ě 0. )!c ` x 4 c, for some x, x ď () c c c 3 c D 3 c c ě 0. c c c c 3 c c 4c 3 4c c ` c 3 4 c ` c c c ď 4 c c c. () From the relations (.6) (.7), after simplifying, we get 4c 3 c 3 4 ` c c x c 4 c x ` 4 c x z, for some x, z with x ď z ď. (3) 3. Main Results We now prove our main result for the function f in the class S pθ, βq.

Mathematics 06, 4, 9 5 of Theorem 3.. Let the function f given by (.) be in the class S pθ, βq. Then $ 6 p βq 4 cos 4 θ & ` 4 a a 4 a 3 3 p βq cos θ, β P 3 ď 3 p βq cos θ ı, β P % p βq cos θ 0, Proof: Let f P S pθ, β; hq g f. From (6) (7) it follows that? cosθ, j? cosθ. " z f e iθ * pzq rp βq p pzq ` βsscosθ ` isinθ (4) f pzq " wg e iθ * pwq rp βq q pwq ` βsscosθ ` isinθ (5) g pwq where p pzq ` c z ` c z ` c 3 z 3 `... P P, pz P q q pwq ` d w ` d w ` d 3 w 3 `... P P, pw P q. Now, equating the coefficients in (4) (5), we have e iθ a c p βq cosθ (6) a e iθ 3 a c p βq cosθ (7) 3a e iθ 4 3a a 3 ` a 3 c 3 p βq cosθ (8) e iθ a d p βq cosθ (9) 3a e iθ a 3 d p βq cosθ (0) 3a e iθ 4 ` a a 3 0a 3 d 3 p βq cosθ () Now from (6) (9) we get c d () c e iθ p p βq cosθ. (3) Now, from (7) (0), we get a 3 e iθ c p βq cos e iθ p βq cosθ pc d q θ `. (4) 4 Additionally, from (8) (), we get a 4 3 e 3iθ c 3 p βq3 cos 3 θ ` 5 8 e iθ c pc d q p βq cos θ ` 6 e iθ pc 3 d 3 q p βq cosθ. (5) Thus, we can easily obtain a a 4 a 3 3 e 4iθ c 4 p βq4 cos 4 θ ` e iθ c 6 pc 3 d 3 q p βq cos θ e 3iθ c 8 pc d q p βq 3 cos 3 θ` e iθ pc d q p βq cos θ 6. (6)

Mathematics 06, 4, 9 6 of According to Lemma (.) Equation (), we get c c ` x `4 c + d d ` x `4 d ñ c d 0 (7) c 3 d 3 c3 `4 c 4 c x c c x (8) a a 4 a 3 e 4iθ c 4 3 p βq4 cos 4 θ ` e iθ c 4 p βq cos θ e iθ c 6 `4 c x p βq cos θ e iθ c `4 c x p βq cos. (9) θ Since p P P, so c ď. Letting c c, we may assume without any restriction that c P r0, s. Thus, applying the triangle inequality on the right-h side of Equation (9), with µ x ď, we obtain a a 4 a 3 ď 3 c4 p βq 4 cos 4 θ ` c4 p βq cos θ ` `4 6 c c µ p βq cos θ ` `4 c c µ p βq cos θ F pµq. Differentiating F pµq, we get `4 F pµq c c p βq cos θ ` c `4 c µ p βq cos θ. (3) 6 Using elementary calculus, one can show that F pµq ą 0 for µ ą 0. This implies that F is an increasing function, it therefore cannot have a maximum value at any point in the interior of the closed region r0, s r0, s. Further, the upper bound for F pµq corresponds to µ, in which case F pµq ď F pq Then 3 c4 p βq 4 cos 4 θ ` c4 p βq cos θ ` 4 c 4 c p βq cos θ G pcq. G pcq ı 3 c p βq cos θ p βq cos θ c `. (3) d Setting G 3 pcq 0, the real critical points are c 0 0, c 0 p βq cos θ. After some calculations we obtain the j following cases: Case : When β P 0,?, we observe that c 0 ě, that is c 0, is out of the interval cosθ p0, q. Therefore, the maximum value of G pcq occurs at c 0 0 or c c 0, which contradicts our assumption of having a maximum value at the interior point of c P r0, s. Since G is an increasing function, the maximum point of G must be on the boundary of c P r0, s, that is c. Thus, we have max G pcq G pq 6 p βq4 cos 4 θ ` 4 0ďcď 3 3 p βq cos θ. (30)

Mathematics 06, 4, 9 7 of Case : When β P? cosθ,, we observe that c 0 ă, that is c 0, is interior of the interval r0, s. Since G pc 0 q ă 0, the maximum value of G pcq occurs at c c 0. Thus, we have g max G pcq G pc 0q G f e 0ďcď 3 3 p βq cos θ This completes the proof of the theorem. Corollary : Let f given by (.) be in the class S σ pβq. Then $ 6 p βq 4 & ` 4 a a 4 a 3 3 p βq, β P 0, 3 ď 3 p βq % p βq ı, β P Corollary : Let f given by (.) be in the class S σ. Then a a 4 a 3 ď 0 3. 3 p βq cos θ p βq cos θ?, j?. These two corollaries coincide with the results of Deniz et al. [8]. Remark 3.: It is observed that for θ 0, we get the Hankel determinant a a 4 a 3 for the class S σ pβq the Hankel determinant of this class was studied by Deniz et al. [8]. 4. Hankel Determinants for the Class of Functions K pθ, β; hq We now estimate an upper bound a a 4 a 3 for the function f pzq in the class K pθ, β; hq. Theorem 4.. Let the f pzq given by (.) be in the class K pθ, β; hq. Then $ & a a 4 a 3 ď % 6 p βq4 cos 4 θ ` 8 6 p βq cos θ, β P 3 p βq cos θ ı, β P p βq cos θ Proof: Let f P K pθ, β; hq g f. From (8) (9) we have 0,? cosθ, j? cosθ " e iθ ` z f * pzq f rp βq p pzq ` βsscosθ ` isinθ (33) pzq " * e iθ ` wg pwq g rp βq p pwq ` βsscosθ ` isinθ (34) pwq where p pzq ` c z ` c z ` c 3 z 3 `..., pz P q q pwq ` d w ` d w ` d 3 w 3 `..., pw P q. Now, equating the coefficients in (33) (34), we have e iθ a c p βq cosθ (35) 6a e iθ 3 4a c p βq cosθ (36) a e iθ 4 8a a 3 ` 8a 3 c 3 p βq cosθ (37) ı.. e iθ a d p βq cosθ (38)

Mathematics 06, 4, 9 8 of 8a e iθ 6a 3 d p βq cosθ (39) e iθ 3a 3 ` 4a a 3 a 4 d 3 p βq cosθ. (40) Now from (35) (38), we get Now, from (36) (39), we get c d (4) e iθ c a p βq cosθ. (4) a 3 e iθ c p ρq cos θ 4 Additionally, from (37) (40), we get e iθ pc d q p ρq cosθ `. (43) a 4 5 48 e 3iθ c 3 p βq3 cos 3 θ ` 5 48 e iθ c pc d q p βq cos θ ` 4 e iθ pc 3 d 3 q p βq cosθ. (44) Thus, we can easily obtain a a 4 a 3 c4 e 4iθ p βq 4 cos 4 θ ` c e 3iθ pc d q p βq 3 cos 3 θ` 96 96 c e iθ pc 3 d 3 q p βq 3 cos 3 e iθ pc d q p βq cos θ θ 48 44. (45) According to Lemma (.), from Equation (4), we get c c ` x `4 c + d d ` x `4 d ñ c d 0 (46) c 3 d 3 c3 `4 c 4 c x c c x a a 4 a 3 c4 e 4iθ p βq 4 cos 4 θ ` c4 e iθ p βq cos θ 96 96 e iθ c `4 c x p βq cos θ e iθ c `4 c x p βq cos θ 48 96 (47). (48) Since p P P, c ď. Letting c c, we may assume without any restriction that c P r0, s. Thus, applying the triangle inequality on the right-h side of Equation (4.6), with µ x ď, we obtain a a 4 a c 4 3 ď e 4iθ p βq 4 cos 4 θ ` c4 e iθ p βq cos θ 96 96 e iθ c `4 c µ p βq cos θ e iθ c `4 c µ p βq cos θ ` 48 96 F pµq. (49) Differentiating F pµq, we get F pµq e iθ c `4 c p βq cos θ 48 e iθ c `4 c µ p βq cos θ `. (50) 48

Mathematics 06, 4, 9 9 of Using elementary calculus, one can show that F pµq ą 0 for µ ą 0. It implies that F is an increasing function it hence cannot have a maximum value at any point in the interior of the closed region r0, s r0, s. Further, the upper bound for F pµq corresponds to µ, in which case `4 F pµq ď F pq ď c4 96 p βq4 cos 4 θ ` c4 96 p βq cos θ ` c c p βq cos θ `4 48 ` c c p βq cos θ G pcq psayq 96. Then G pcq c3 4 p βq4 cos 4 θ ` c3 8c 4c 4 p βq cos 3 p βq cos θ θ `. (5) 3 g Setting G pcq 0, the real critical points are c 0 0, c 0 f e 6 p βq cos θ After some calculations we obtain the following cases: Case : When β P 0,? j, we observe that c 0 ě, that is c 0, is out of the interval cosθ p0, q. Therefore, the maximum value of G pcq occurs at c 0 0 or c c 0, which contradicts our assumption of having the maximum value at the interior point of c P r0, s. Since G is an increasing function, the maximum point of G must be on the boundary of c P r0, s, that is c. Thus, we have Case : When β P max G pcq G pq 0ďcď 6 p βq4 cos 4 θ ` 6 p βq cos θ.?,, we observe that c 0 ă, that is c 0, is interior of the interval cosθ r0, s. Since G pc 0 q ă 0, the maximum value of G pcq occurs at c c 0. Thus, we have g max G pcq G pc 0q G f 6 e ı 0ďcď p βq cos θ. 3 p βq cos θ ı 8 p βq cos θ This completes the proof of the theorem. Corollary : Let f given by () be in the class K σ pβq. Then $ p βq 4 p βq & `, β P 0,? a a 4 a 6 6 j 3 ď 3 p βq % 8 p βq ı, β P?., Corollary : Let f given by () be in the class K σ. Then a a 4 a 3 ď 3. These two corollaries coincide with the results of Deniz et al. [8]. ı.

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