Báo cáo sinh học: " Research Article On Certain Multivalent Starlike or Convex Functions with Negative Coefficients" pdf

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 751721, 11 pages doi:10.1155/2010/751721 Research Article On Certain Multivalent Starlike or Convex Functions with Negative Coefficients ˇ Neslihan Uyanik,1 Erhan Deniz,2 Ekrem Kadioglu,2 and Shigeyoshi Owa3 Department of Mathematics, Kazim Karabekir Faculty of Education, Ataturk University, ă Erzurum 25240, Turkey Department of Mathematics, Science and Art Faculty, Ataturk University, Erzurum 25240, Turkey ă Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan Correspondence should be addressed to Shigeyoshi Owa, shige21@ican.zaq.ne.jp Received April 2010; Accepted June 2010 Academic Editor: N Govil Copyright q 2010 Neslihan Uyanik et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited By means of a differential operator, we introduce and investigate some new subclasses of pvalently analytic functions with negative coefficients, which are starlike or convex of complex order Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out Introduction Let Am p denote the class of functions of the following form: f z ∞ zp − ak zk ak ≥ 0; m, p ∈ N : {1, 2, 3, } , 1.1 k p m which are analytic and multivalent in the open unit dics U {z : z ∈ C and |z| < 1} Let f q denote the qth-order ordinary differential operator for a function f ∈ Am p , that is, f where p > q; p ∈ N; q ∈ N0 q z p! zp−q − p−q ! k N ∪ {0}, z ∈ U ∞ p m k! ak zk−q , k−q ! 1.2 Journal of Inequalities and Applications q as k−q k! k−q ! p−q n Next, we define the differential operator Dn f Dn f q p! zp−q − p−q ! k z ∞ p m m ∈ N; z ∈ U ak zk−q 1.3 In view of 1.3 , it is clear that D0 f q z f q D1 f z, n D f q q z Df q Df q z D n−1 z f p−q z q z , 1.4 z If we take p and q for Dn f q , then Dn f q become the differential operator defined by S˘ l˘ gean aa n Finally, in terms of a differential operator Dn f q defined by 1.3 above, let Em,p q denote the subclass of Am p consisting of functions f which satisfy the following inequality: n Em,p ⎧ ⎨ q f ∈ Am ⎩ Dn f q z p : / 0, z ∈ C − {0} , f z zp−q ∞ z − p k p ⎫ ⎬ ak z , ak ≥ , ⎭ m k 1.5 where k ∈ N, n ∈ N0 , k > n, m ∈ N; k − q / p − q ≥ p − q − n − ≥ 0; z ∈ U n For m ∈ N, n ∈ N0 , and γ ∈ C − {0}, we define the next subclasses of Em,p q n Em,p q, γ n Nm,p q, γ n f ∈ Em,p q : Re ⎧ ⎨ ⎩ k p m ≤ n Km,p q, γ ∞ n f ∈ Em,p q : ⎧ ⎨ p! p−q ! n f ∈ Em,p q : ⎩ ≤ ∞ k p m p! γ −p p−q ! q Dn f q z q n n n Re γ γ k−q k! k−q ! p−q q −p z k−q k! k−q ! p−q −p γ Dn f γ n q > 0, z ∈ U n k−q −p p−q q n Re γ γ , γ ak , k−q −p p−q q n γ ak , 1.6 where γ ∈ C − {0}; m ∈ N; k − q / p − q ≥ p − q − n − ≥ 0; z ∈ U Journal of Inequalities and Applications S∗ b was studied by Nasr and Aouf also see Bulboac˘ et al a Remark 1.1 Em,1 0, γ Tα m and Em,1 0, − α Cα m , α ∈ 0, were introduced by Em,1 0, − α Srivastava et al T ∗ α and E1,1 0, − α C α , α ∈ 0, were introduced by E1,1 0, − α Silverman 1 Om γ and Km,1 0, γ Om γ were introduced by Parvathan and Km,1 0, γ Ponnusanny 6, pages 163-164 n n n For p and q 0, the classes Em,p q, γ , Nm,p q, γ , and Km,p q, γ are closely aa related with Tn,m γ , On,m γ , and Pn,m γ which are defined by Owa and S˘ l˘ gean in n n In this paper we give relationships between the classes of Em,p q, γ , Nm,p q, γ , and q, γ In the particular case when m ∈ N and n 0, p 1, and q 0, we obtain the same results as in n Km,p Main Results Our main results are contained in Theorem 2.1 Let m ∈ N, n ∈ N0 and let γ ∈ C − {0}; then n n Km,p q, γ ⊆ Em,p q, γ ; n n Em,p q, γ ⊆ Nm,p q, γ ; if γ ∈ 0, ∞ , then n Km,p q, γ n Em,p q, γ n Nm,p q, γ ; 2.1 n if γ ∈ −∞, , then Nm,p q, γ ⊆ Em,p q, γ ; / n n if γ ∈ −∞, , then Em,p q, γ / Km,p q, γ ⊆ n n Proof Let f ∈ Km,p q, γ We prove that Dn f q Dn f q z z −p n < γ , q z∈U 2.2 If f has the series expansion f z zp − ∞ ak zk , k p m ak ≥ 0, 2.3 Journal of Inequalities and Applications then Dn f Dn f ≤ q z q −p z n − γ q − p!/ p − q ! − p ∞ k p m p!/ p − q ! − ∞ k p m k!/ k − q ! k!/ k − q ! q k−q / p−q n k−q / p−q ∞ k p m p!/ p − q ! − n ak k!/ k − q ! n ak |z|k−p k−q / p−q −p k−q / p−q n n |z|k−p q ak |z|k−p − γ 2.4 We use the fact that Dn f p!/ p − q !; these imply q z /zp−q / for z ∈ U − {0} and limz → o Dn f p! − p−q ! k ∞ p m n k−q k! k−q ! p−q q z /zp−q ak |z|k−p > 2.5 for z ∈ U From 2.4 and 2.5 , we deduce Dn f q Dn f q < z z −p n − γ q − p!/ p − q ! − p ∞ k p m p!/ p − q ! − ∞ k p m q γ n k!/ k − q ! k − q / p − q k!/ k − q ! k−q / p−q ∞ k p m p!/ p − q ! − n ak n ak k−q / p−q −p k!/ k − q ! k−q / p−q q n ak n γ 2.6 n By using the definition of Km,p q, γ from this last inequality we, obtain 2.2 which implies Re n hence f ∈ Em,p q, γ γ Dn f Dn f q q z z −p q n > −1 z∈U , 2.7 Journal of Inequalities and Applications n Let f be in Em,p q, γ Then 2.7 holds and, by using 2.3 , this is equivalent to ⎧ ⎛ ⎨1 p!/ p − q ! zp−q − ⎝ Re ⎩γ p!/ p − q ! zp−q − > −1 ∞ k p m k!/ k − q ! ∞ k p m k!/ k − q ! k−q / p−q k−q / p−q n n ak zk−q ak zk−q −p q ⎞⎫ ⎬ n⎠ ⎭ z∈U 2.8 t ∈ 0, if t → 1− , from 2.8 we obtain For z ⎧⎛ ⎨ − p!/ p − q ! ⎝ ⎩ p!/ p − q ! − ∞ k p m k!/ k − q ! ∞ k p m k!/ k − q ! n k−q / p−q k−q / p−q n ak ak ⎞⎫ ⎬ γ ⎠ ≤ −p ⎭ Re γ q n 2.9 which is equivalent to ∞ k p m k−q k! k−q ! p−q n k−q p−q γ −p Re γ n ak ≤ q p! p−q ! γ −p Re γ q n 2.10 n Then multiplying the relation last inequality with Re γ/|γ|, we obtain f ∈ Nm,p q, γ n n if γ is a real positive number, then the definitions of Nm,p q, γ and Km,p q, γ are n n Km,p q, γ By using and from this theorem, we obtain equivalent, hence Nm,p q, γ We have the following two cases Case γ ∈ p − q − n − − m/ p − q , Let fm,α p, q, n; z be defined by zp − α fm,α p, q, n; z m−q p−q p −n p m−q ! p p! z p m ! p−q ! m 2.11 and let α > We have ∞ k p m n k−q k! k−q ! p−q p ≤ p m−q ! p ×α α m ! p m−q p−q p! p−q ! p k−q −p p−q m−q p−q −n q n p n Re γ γ m−q −p p−q p m−q ! p! p m ! p−q ! m−q −p p−q q n γ −γ −γ γ q ak n Re γ γ γ 2.12 Journal of Inequalities and Applications or ∞ k p m ≤ −α −p n and then fm,α p, q, n; z ∈ Nm,p q, γ Let now k−q −p p−q p! −p p−q ! p! p−q ! < F z n k−q k! k−q ! p−q q q n n Re γ γ Re γ γ n m p−q 1 q γ ak ≤0 γ 2.13 , γ n see the definition of Nm,p q, γ ⎛ q ⎝ Dn fm,α p, q, n; z −p q γ Dn f p, q, n; z ⎞ n⎠ q z∈U 2.14 m,α Then, by a simple computation and by using the fact that q fm,α p, q, n; z q fm,α z −n p m ! p! p m−q zp−q − α p−q p−q ! p m−q ! p! p! p m−q zp−q − α p−q p−q ! p−q ! q Dn fm,α z −n p! p m−q p! zp−q − α p−q p−q ! p−q ! −n p zp m−q p p m−q ! p p! z m ! p−q ! m−q m−q p−q n zp m−q p! zp−q − αzm , p−q ! q Dn fm,α z p m−q m p! z , zp−q − α p−q p−q ! 2.15 Journal of Inequalities and Applications we obtain q Dn fm,α z 1 γ F z γ p!/ p − q ! zp−q − α p −p q q Dn fm,α z q n m − q / p − q zm p!/ p − q ! zp−q − αzm n where ζ −p a − αbζ γ − αζ zm , a −p q − αzm −p q n γ − αzm n n 2.16 m−q / p−q q m/ p − q , and ϕ ζ , 1, b −p ϕζ For α > we, have ϕ U p −p q n a − αbζ γ − αζ 2.17 C∞ − D c, d , where D is the disc with the center c α2 b − a γ α2 − 2.18 d α b−a γ − α2 2.19 and the radius We have F U C∞ − D c 1, d where D c, d {w : |w − c| < d} and we deduce that Re F z > for all z ∈ U does not hold n ∈ n We have obtained that for α > 1, fm,α ∈ Nm,p q, γ , but fm,α / Em,p q, γ and in this case n n ⊆ Nm,p q, γ / Em,p q, γ Case γ ∈ −∞, p − q − n − − m/ p − q We consider the function fm,α defined by 2.11 for α ∈ 1, −p q n γ / −p q n m/ p − q In this case, the inequality 2.13 holds too and this implies that n fm,α ∈ Nm,p q, γ We also obtain that f / Em,p q, γ like in Case ∈ n Journal of Inequalities and Applications Let f fm,α be given by 2.11 , where α > |γ| − p m/ p − q and |γ| − p q n m/ p − q > Then ∞ n k−q k! k−q ! p−q k p m p ×α > p m−q ! p α m ! p p m−q p−q −n m−q p−q p! p−q ! k−q −p p−q n −p p n n / |γ| − p q n q n q n q γ ak m−q −p p−q γ p m−q ! p! · p m ! p−q ! m−q −p p−q p! γ p−q ! q q 2.20 γ n which implies that fm,α / Km,p q, γ ∈ n for m ∈ N, n ∈ N0 , γ ∈ −∞, 2.21 We have F z 1 γ q Dn fm,α z q Dn fm,α z −p q n a − αbζ γ − αζ ϕζ , 2.22 where ϕ is given by 2.17 From ϕ U D c, d where c and d are given by 2.18 and 2.19 , we obtain Re F z ≥ If γ ∈ −∞, p − q − n − − m/ p − q m/ p − q , , then α γ b γ α αb γ α and α ∈ γ a a |γ| − p > 0, 2.23 q n / |γ| − p q n 2.24 Journal of Inequalities and Applications and if γ∈ α∈ γ −p γ −p q q n n m ,0 , p−q p−q−n−1− γ −p , m/ p − q −p n q q m/ p − q n − γ ∩ 0, , 2.25 n then 2.24 also holds By combining 2.24 with 2.23 and the definition of Em,p q, γ , we obtain that n fm,α ∈ Em,p q, γ for α ∈ γ −p γ −p q q n n m/ p − q , γ −p −p q n m/ p − q − γ q n ∩ 0, , γ ∈ −∞, 2.26 Appendix n In this paper, we discuss the class Em,p q, γ of analytic functions with negative coefficients Let us consider the functions f given by ∞ zp f z ak zk A.1 k p which are analytic in U For such a function f, we say that f ∈ Gn q, γ if it satisfies 1,p Re 1 γ Dn f q Dn f q z z −p q n >0 z∈U A.2 for some complex number γ with < Re 1/γ < 1/ p − q − n − If we define the function F for f ∈ Gn q, γ by 1,p F z 1/γ Dn f q z /Dn f q z −p 1−p q q n −i 1−p n Re 1/γ q n Im 1/γ , A.3 10 Journal of Inequalities and Applications then we know that F is analytic in U, F 1, and Re f z > z ∈ U Thus F is the Carath´ odory function Since the extremal function for the Carath´ odory function F is given e e by z , 1−z F z A.4 we can write 1/γ Dn f q z /Dn f z −p q 1−p q q n −i 1−p q n Im 1/γ z 1−z n Re 1/γ A.5 This shows us that Dn f q Dn f q z z −p q γ − iγ − p n q γ n Im 1−p γ q n Re γ z 1−z A.6 Noting that Dn f q z Dn f p−q z q z , A.7 iγ − p q 1−z , z we see that Dn f q z p − q Dn f q z γ − p−q−n−γ z 1−p q n Re γ γ n Im A.8 that is, Dn f q z p − q Dn f q z z − 1−p 2γ q γ n Re 1−z A.9 It follows from the above that z Dn f q t p − q Dn f q t − t dt 2γ 1−p q γ n Re z dt 1−t A.10 Calculating the above integrations, we have that log Dn f p−q q z − log z −2γ 1−p q n Re γ log − z A.11 Journal of Inequalities and Applications 11 Therefore, we obtain that Dn f q z z 1/ p−q 1−z 2γ , 1−p q n Re 1/γ A.12 that is, n D f q z z 1−z 2γ 1−p q n Re 1/γ p−q A.13 Consequently, the function f defined by the above is the extremal function for the class n Gn q, γ But our class Em,p q, γ is defined with analytic functions f with negative 1,p coefficients Thus we not know how we can consider the extremal function for this class References ˇ aa G S S˘ l˘ gean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar Part (Bucharest, 1981), vol 1013 of Lecture Notes in Mathematics, pp 362–372, Springer, 1983 M A Nasr and M K Aouf, “Starlike function of complex order,” The Journal of Natural Sciences and Mathematics, vol 25, no 1, pp 1–12, 1985 T Bulboac˘ , M A Nasr, and G S S˘ l˘ gean, “Function with negative coefficients n-starlike of complex a ¸ aa order,” Universitatis Babes-Bolyai Studia Mathematica, vol 36, no 2, pp 7–12, 1991 ¸ H M Srivastava, S Owa, and S K Chatterjea, “A note on certain classes of starlike functions,” Rendiconti del Seminario Matematico della Universit` di Padova, vol 77, pp 115–124, 1987 a H Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol 51, pp 109–116, 1975 R Parvathan and S Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New Trends in Geometric Function Theory and Applications S Owa and G S S˘ l˘ gean, “Starlike or convex of complex order functions with negative coefficients,” aa Surikaisekikenkyusho K¯ kyuroku, no 1062, pp 77–83, 1998 o ¯ ¯ ¯ S Owa and G S S˘ l˘ gean, “On an open problem of S Owa,” Journal of Mathematical Analysis and aa Applications, vol 218, no 2, pp 453–457, 1998  ... S Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New Trends in Geometric Function Theory and Applications S Owa and G S S˘ l˘ gean, ? ?Starlike or convex of complex order... 1/γ p−q A.13 Consequently, the function f defined by the above is the extremal function for the class n Gn q, γ But our class Em,p q, γ is defined with analytic functions f with negative 1,p coefficients... certain classes of starlike functions, ” Rendiconti del Seminario Matematico della Universit` di Padova, vol 77, pp 115–124, 1987 a H Silverman, “Univalent functions with negative coefficients,” Proceedings