Coefficient Estimates for Bazilevič Ma-Minda Functions in the Space of Sigmoid Function

Transcription

1 Malaya J. Mat. 43)206) Coefficient Estimates for Bazilevič Ma-Minda Functions in the Space of Sigmoid Function Sunday Oluwafemi Olatunji a, and Emmanuel Jesuyon Dansu b a,b Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria. Abstract In this work, the authors investigated the coefficient estimates for Bazilevič Ma-Minda Functions for the class T α n λ, β, l, Φ). The first few coefficient bounds for this class were obtained and also the relevant connection to Fekete-Szegö theorem and were briefly discussed. Our results serve as a new generalization in this direction and gives birth to many corollaries. Keywords: Analytic Function, Univalent Function, Starlike Function, Convex Function, Bazilevič Function, Subordination, Sigmoid Function, Fekete-Szegö Inequality. 200 MSC: 30C45, 33E99. c 202 MJM. All rights reserved. Introduction In the twentieth century, the theory of special functions was overshadowed by other fields like functional analysis, real analysis, algebra, topology, differential equations and so on. These functions do not have specific definitions but they constitute an information process that is inspired by the way biological nervous system such as the brain processes information. This information process contains large numbers of highly interconnected elements neurons) working together to perform specific tasks. Special functions can be categorized into three, namely ramp function, sigmoid function and threshold function. The most popular of the functions is the sigmoid function because of its gradient descent algorithm. It can be evaluated by truncated series expansion see details in 5], 9] and ]). The sigmoid function of the form is differentiable and has the following properties: i) it outputs real numbers between 0 and. gz) = ii) it maps a very large input domain to a small range of outputs. iii) it never loses information because it is a one-to-one function. iv) it increases monotonically. + e z.) The four properties show that sigmoid function is very useful in geometric functions theory. Corresponding author. olatunjiso@futa.edu.ng S.O. Olatunji) and ejdansu@futa.edu.ng E.J. Dansu).

2 506 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions... Let A denote the class of functions of the form f z) = z + a k z k z U).2) which are analytic in the open disk U = {z : z < } and normalized by f 0) = f 0) = 0. A domain U C is convex if the line segment joining any two points in U lies entirely in U, while a domain is starlike with respect to a point ω 0 U if the line segment joining any point of U to ω 0 lies inside U. A function f A is starlike if f U) is a starlike domain with respect to the origin and convex if f U) is convex. Recall that starlike and convex ) functions are denoted by ST and CV respectively and analytically written as Re z f z) > 0 and Re + z f z) f z) f > 0. Starlike and convex functions of type α are denoted by STα) and z) ) CVα) respectively and characterized by Re z f z) > α and Re + z f z) f z) f > α where α : 0 α < see detail z) in 2]). The two functions f and g are analytic in the open unit disk U. We say f is subordinate to g written as f < g U if there exists a Schwarz function wz) which is analytic in U with w0) = 0 and wz) < such that f z) = gwz)). It follows from Schwarz lemma that f z) < gz) z U) = f 0) = g0) and f U) gu) see details in 8]). Ma and Minda7] unified various subclasses of starlike and convex functions for which either of the quantity z f z) or + z f z) f z) f is subordinate to a more general superordinate function. For this purpose, they z) considered an analytic function ϕ with positive real part in the open unit disk U, ϕ0) = and ϕ 0) > 0 and ϕ maps U onto a region starlike with respect to and symmetric with respect to the real axis. The class of Ma- Minda starlike function consists of functions f A satisfying the subordination z f z) < ϕz) and Ma-Minda f z) convex function consists of functions f A satisfying subordination + z f z) f < ϕz) detail in 2]). z) Lemma. Pommerenke3]). If a function p P is given by pz) = + p z + p 2 z z U).3) then p k 2 k N), where P is the class of Caratheodory function, analytic in U for which p0) = and Re pz) > 0 z U). Let α > 0 α is real), then which gives Or, equivalently f z) α = z + Using simple expansion for.6), we have f z) α = z α + αa 2 z + a 3 z 2 + a 4 z ) + a k z k ) α.4) f z) α = z + a 2 z 2 + a 3 z 3 + a 4 z ) α.5) f z) α = z + a 2 z + a 3 z 2 + a 4 z )) α.6) Since the expansion continues, then ) f z) α = z α + αa 2 z + a 3 z 2 + a 4 z ) ) αα ) a 2 z + a 3 z 2 + a 2! 4 z ) ) which implies f z) α = z α + αa 2 z α+ + αa 3 z α+2 + αa 4 z α

3 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions This finally gives f z) α = z α + Catas et al.3] defined the Catas Operator as follows: I 0 λ, l) : A A I 0 λ, l) f z) = f z) ) ) λ + l λz I λ, l) f z) = Iλ, l) f z)) + Iλ, l) f z)) + l + l and ) ) λ + l λz I 2 λ, l) f z) = I λ, l) f z)) + I λ, l) f z)) + l + l In general, I n λ, l) f z) = Iλ, l)i n λ, l) f z)) = z + Applying.9) in.8), we have ) + λα ) + l n I n λ, l) f z) α = z α + + l where n N 0, α > 0 α is real), λ 0, l 0. a k α)z α+k.8) = z + = z + + λk ) + l + l ) a k z k ) + λk ) + l 2 a k z k + l ) + λk ) + l n a k z k.9) + l ) + λα + k 2) + l n a k α)z α+k.0) Oladipo and Olatunji0] used.0) to define a class T α n λ, β, l) with geometric condition satisfying + l Re I n λ, l) f z) α z α +λα )+l +l > β.) where n N 0, α > 0 α is real), λ 0, l 0 and 0 β <. The first few coefficient bounds for the class were obtained and the coefficient inequalities for the class were derived by employing Hayami s method 6]. By specializing the parameters involved in.), we obtain various subclasses of analytic functions studied by ], 2], 4], 5] and so on. In this work, the authors defined a new class of functions denoted by T α n λ, β, l, Φ) as related to modified sigmoid function with geometric condition satisfying Re I n λ,l) f z) α +λα )+l +l β z α β < ϕz).2) where n N 0, α > 0 α is real), λ 0, l 0 and 0 β <. The first few coefficient estimates for the class are obtained. Also, the relevant connection to Fekete-Szegö theorem are briefly discussed. For the purpose of our results, we require the following lemmas. Lemma.2 Fadipe-Joseph et al.5]). Let g be a sigmoid function and Φz) = 2gz) = + m= ) m 2 m n= then Φz) P, z < where Φz) is a modified sigmoid function. Lemma.3 Fadipe-Joseph et al.5]). Let then Φ m,n z) < 2. Φ m,n z) = 2gz) = + m= ) m 2 m n= ) m z n.3) n! ) z n n!.4)

4 508 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions... Lemma.4 Fadipe-Joseph et al.5]). If Φz) P and it is starlike, then f is a normalized univalent function of the form.2). Setting m =, Fadipe-Joseph et al.5] remarked that Φz) = + n= c nz n where c n = )n+. As such, 2n)! c n 2, n =, 2, 3,... and the result is sharp for each n. 2 Coefficient Estimates In the sequel, it is assumed that ϕ is an analytic function with positive real part in the open unit disk U, with ϕ0) =, ϕ 0) > 0 and ϕu) is symmetric with respect to the real axis. Such a function has a series expansion of the form ϕz) = + β z + β 2 z 2 + β 3 z 3 + β 4 z β > 0) 2.5) For functions in the class T α n λ, β, l, Φ), the following results are obtained. Theorem 2.. If f z) α T α n λ, β, l, Φ) is given by.2), then a 3 α) a 2 α) β) 2αB 2 B ) +λα+l +λα )+l α 2 +λα+l β)b 2.6) 4α +λα+l +λα )+l +λα )+l ) ] α ) β)b 2 +λα+)+l n +λα )+l +λα+)+l +λα )+l 2.7) a 4 α) 2 β)3b 3 6B 2 B ) ) +λα+2)+l n +λα )+l α ) β)3 B ) 3n 3 +λα+l +λα )+l 3 2αB 2 B ) +λα+l +λα )+l ) ] α ) β)b 2 +λα+)+l n ) +λα+)+l n +λα )+l +λα )+l + α 2) β)2 B 3 2.8) Proof. Let f z) α T α n λ, β, l, Φ). Then there are analytic functions u : U U with u0) = 0 satisfying I n λ,l) f z) α +λα )+l +l β z α β = ϕuz)) 2.9) Define the function Φz) by or, equivalently Φz) = + uz) uz) = + 2 z 24 z z5 64 z z ) uz) = Φz) Φz) + = 4 z 6 z2 92 z z z ) In view of 2.9), 2.20) and 2.2), clearly I n λ,l) f z) α +λα )+l +l β z α β = ϕ ) Φz) Φz) ) Using 2.2) together with 2.5), it is evident that ) Φz) ϕ = + B Φz) + 4 z + B 2 B 6 z 2 B + 6B 2 3B 3 92 z 3 + 5B + B 2 9B 3 + 3B 4 z ) 768

5 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions Recall that which has the expansion + λα + l + α + λα ) + l + αa 4 + αα )a 2 a αa α I n λ, l) f z) α = + z α +λα )+l +l αα ) 2a 2 a 2! 4 + a 2 3 Therefore 2.22) yields + λα + l + λα ) + l + αa 4 + αα )a 2 a αa 5 + αα ) 2! +... = β + β) a 2 z + αa 3 + αα )α 2) a αα ) 2 a 2 2 ) + λα + k 2) + l n a + λα ) + l k α)z k ) ) + λα + ) + l n z 2 + λα ) + l z 3 ) + λα + 2) + l + λα ) + l αα )α 2) ) + a a 3 + a 2 z + αa 3 + 2a 2 a 4 + a B αα )α 2) a αα ) 2 a 2 2 αα )α 2)α 3) a ) ) + λα + ) + l n z 2 + λα ) + l z 3 ) + λα + 2) + l + λα ) + l αα )α 2) ) + a a z + B 2 β 6 z 2 B + 6B 2 3B 3 92 αα )α 2)α 3) a Comparing the L.H.S. and R.H.S. of 2.25), it gives ) + λα + l n α a + λα ) + 2 α) = β)b l 4 αa 3 α) + αa 4 α) + αα )a 2 α)a 3 α) + So, by simple computation, we obtain z 3 + 5B + B 2 9B 3 + 3B 4 z ) ) αα ) + λα + ) + l n a α) = β)b 2 B ) + λα ) + l 6 ) ) + λα + 3) + l n z 4 + λα ) + l 2.24) ) ) + λα + 3) + l n z 4 + λα ) + l ] 2.25) 2.26) 2.27) ) ) αα )α 2) + λα + 2) + l n a α) = β)b + 6B 2 3B 3 ) + λα ) + l ) a 3 α) a 2 α) β) 2αB 2 B ) +λα+l +λα )+l α 2 +λα+l β)b 4α +λα+l +λα )+l +λα )+l 2.29) ) ] α ) β)b 2 +λα+)+l n +λα )+l +λα+)+l +λα )+l 2.30) a 4 α) 2 β)3b 3 6B 2 B ) ) +λα+2)+l n +λα )+l α ) β)3 B ) 3n 3 +λα+l +λα )+l 3 2αB 2 B ) +λα+l +λα )+l ) ] α ) β)b 2 +λα+)+l n ) +λα+)+l n +λα )+l +λα )+l + α 2) β)2 B 3 2.3)

6 50 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions... and this completes the proof of Theorem 2.). By specializing some parameters that are involved, we obtain some corollaries. Setting β = 0, it gives the following corollary Corollary 2.. If f z) α Tn α λ, 0, l, Φ) is given by.2), then a 3 α) a 2 α) 2αB 2 B ) +λα+l +λα )+l α 2 B 2.) 4α +λα+l +λα )+l +λα+l +λα )+l ) ] α )B 2 +λα+)+l n +λα )+l +λα+)+l +λα )+l 2.33) a 4 α) 23B 3 6B 2 B ) +λα+2)+l +λα )+l α )B 3 ) 3n +λα+l +λα )+l 3 2αB 2 B ) +λα+l ) ] α )B 2 +λα+)+l n +λα )+l ) +λα+)+l n +λα )+l +λα )+l 2.34) + α 2)B3. Setting α = in Corollary 2.) gives Corollary 2.2. If f z) T nλ, 0, l, Φ) is given by.2), then Putting λ = in Corollary 2.2) yields a 3 ) a 2 ) B 4 +λ+l +l 2B 2 B ) +λ+l +l +λ+l +l Corollary 2.3. If f z) T n, 0, l, Φ) is given by.2), then 2.35) +2λ+l +l ] 2.36) a 4 ) 23B 3 6B 2 B ) B λ+l ) +l a 3 ) Taking l = 0 in Corollary 2.3) it is seen that a 2 ) B 2.38) 4 2+l +l 2B 2 B ) 2+l +l 2+l +l 3+l +l ] 2.39) a 4 ) 23B 3 6B 2 B ) B l ) +l

7 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions... 5 Corollary 2.4. If f z) T n, 0, 0, Φ) is given by.2), then a 2 ) B ) a 3 ) 2B2 B )2] 2 3 n 2.42) If n = 0 in Corollary 2.4) we get Corollary 2.5. If f z) T0, 0, 0, Φ) is given by.2), then a 4 ) 23B 3 6B 2 B ) 3844 B ) a 2 ) B 4 a 3 ) B 2 B ) ) 2.45) 3 The Fekete-Szegö Inequality a 4 ) 3B 3 6B 2 B ) 92 B ) In order to obtain the Fekete-Szegö Inequalities, we shall employ the Deniz and Orhan4] and Ma and Minda7] approach. Theorem 3.. If f z) α Tn α λ, β, l, Φ) is given by.2), then a 3 µa 2 2 β B 2β )α + 2µ ) +λα+)+l +λα )+l α 2 +λα+)+l +λα )+l Proof. From 2.29) and 2.30), we have a 3 µa 2 2 = β) 2αB 2 B ) +λα+l +λα )+l α 2 +λα+l +λα )+l 2αB B 2 ) +λα+l +λα+l +λα )+l +λα )+l ) ] α ) β)b 2 +λα+)+l n +λα )+l µ +λα+)+l +λα )+l Simplifying 3.48), we have a 3 µa 2 2 = β B 2β )α + 2µ ) +λα+)+l +λα )+l which completes the proof. Taking µ =, we obtain α 2 +λα+)+l +λα )+l Corollary 3.6. If f z) α Tn α λ, β, l, Φ) is given by.2), then a 3 a 2 2 β B 2β )α + ) +λα+)+l +λα )+l α 2 +λα+)+l +λα )+l 4 Conclusion 2αB B 2 ) +λα+l +λα+l +λα )+l +λα )+l 2αB B 2 ) +λα+l +λα+l +λα )+l +λα )+l By varying other parameters that are involved, many corollaries can be generated ) β)b 4α +λα+l +λα )+l 3.48) ). 3.50)

8 52 S.O. Olatunji and E.J. Dansu. / Coefficient Estimates for Bazilevič Ma-Minda Functions... 5 Acknowledgement The authors would like to thank the anonymous reviewers for their valuable comments. References ] S. Abdulhalim, On a Class of Analytic Functions Involving Salagean Differential Operator, Tamkang Journal of Mathematics, 23) 992), ] R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramaniam, Coefficient Estimates for Bi-Univalent Ma- Minda Starlike and Convex Functions, arxiv v mathcv], 20 Aug 20). 3] A. Catas, G.I. Oros and G. Oros, Differential Subordinations Associated with Multiplier Transporters, Abstract Appl. Anal., ID ), -. 4] E. Deniz and H. Orhan, The Fekete-Szegö Problem for a Generalized Subclass of Analytic Functions, Kyungpook Math. J., ), ] O.A. Fadipe-Joseph, A.T. Oladipo and U.A. Ezeafulukwe, Modified Sigmoid Function in Univalent Function Theory, International Journal of Mathematical Sciences and Engineering Applications, 77) 203), ] T. Hayami, S. Owa and H.M. Srivastava, Coefficient Inequalities for Certain Classes of Analytic and Univalent Functions, Journal of Inequalities in Pure and Applied Math., Art. 95, 84) 2007), -2. 7] W.C. Ma and D. Minda, A Unified Treatment of Some Special Cases of Univalent Functions, In Proceedings of the Conference on Complex Analysis Tianjin), Conference Proceedings, Lecture Notes Anal. I, International Press, Cambridge, MA, 992), ] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications, Series of Monographs and Text Books in Pure and Applied Mathematics, Marcel Dekker, New York, ). 9] G. Murugusundaramoorthy and T. Janani, Sigmoid Function in the Space of Univalent λ-pseudo Starlike Functions, International Journal of Pure and Applied Mathematical Sciences, 0 ) 205), ] A.T. Oladipo and S.O. Olatunji, On a Certain Subclass of Bazilevic Functions Defined by Catas Operator, International Journal of Mathematical Sciences and Application, ) 20), 9. ] S.O. Olatunji, A.M. Gbolagade, T. Anake and O.A. Fadipe-Joseph, Sigmoid Function in the Space of Univalent Function of Bazilevič Type, Scientia Magna, 973) 203), ] T.O. Opoola, On a New Subclass of Univalent Functions, Mathematicae Cluj 36), 592) 994), ] C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 975). 4] R. Singh, On Bazilevic Functions, Proceedings of the American Mathematical Society, 38, 2) 973), ] K. Yamaguchi, On Functions Satisfying Re f z) z MR33-356, 7 966), > 0, Proceedings of the American Mathematical Society, Received: September 0, 205; Accepted: July 7, 206 UNIVERSITY PRESS Website: