A Note on Coefficient Inequalities for (j, i)-symmetrical Functions with Conic Regions

Transcription

1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) ISSN (o) wwwimviblorg /JOURNALS / BULLETIN Vol 6(2016) Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o) ISSN X (p) A Note on Coefficient Inequalities for (j i)-symmetrical Functions with Conic Regions Fuad S M Al Sarari and SLatha Abstract In this note the concepts of (j i)-symmetric points Janowski functions and the conic regions are combined to define a class of functions in a new interesting domain which represents the conic type regions Certain interesting coefficient inequalities are deduced 1 Introduction Let A denote the class of functions of form (11) f(z) = z + a n z n which are analytic in the open unit disk U = {z : z C and z < 1} and S denote the subclass of A consisting of all function which are univalent in U Given two functions f and g analytic in U we say that the function f is subordinate to g in U and write f(z) g(z) if there exists a Schwarz function w which is analytic in U with w(0) = 0 and w(z) < 1 such that f(z) = g(w(z)) z U If g is univalent in U then f g if and only if f(0) = g(0) and f(u) g(u) Using the principle of the subordination we define the class P of functions with positive real part see [2] Definition 11 Let P denote the class of analytic functions of the form p(z) = 1 + n=1 p nz n defined on U and satisfying p(0) = 1 R{p(z)} > 0 z U Any function p in P has the representation p(z) = 1+w(z) 1 w(z) where w(0) = 0 w(z) < 1 on U The class of functions with positive real part P plays a crucial role in geometric function theory Its significance can be seen from the fact 2010 Mathematics Subject Classification 30C45 Key words and phrases Analytic functions (j i)-symmetric points Conic domains Janowski functions k-starlike functions k- Uniformaly convex functions 77

2 78 AL SARARI AND LATHA that simple subclasses like class of starlike S class of convex functions C class of starlike functions with respect to symmetric points have been defined by using the concept of class of functions with positive real part Definition 12 [1] Let P[A B] where 1 B < A 1 denote the class of analytic function p defined on U with the representation p(z) = 1+Aw(z) 1+Bw(z) z U w(0) = 0 w(z) < 1 p P[A B] if and only if p(z) 1+Az 1+Bz Geometrically a function p(z) P[A B] maps the open unit onto the disk defined by the domain { Ω[A B] = w : w 1 AB 1 B 2 < A B } 1 B 2 The class P [A B] is connected the class P of function with positive real parts by the relation (A + 1)p(z) (a 1) p(z) P P[A B] (B + 1)p(z) (B 1) This class was introduced by Janowski [1] and then studied by several authors Kanas and Wisniowska [3 8] introduced and studied the class k UCV of k- uniformly convex functions and the corresponding class k ST of k-starlike functions These classes were defined subject to the conic region Ω k k 0 [3 8] as Ω k = {u + iv : u > k (u 1) 2 + v 2 } This domain represents the right half plane for k = 0 hyperbola for 0 < k < 1 a parabola for k = 1 and ellipse for k > 1 The functions p k (z) which play the role of extremal functions for these conic regions are given as 1+z 1 z k ( = 0 ) π log 1+ z (12) p k (z) = 2 1 z k = k sinh 2 [( 2 2 π arccos k) arctan h z ] [ ] 0 < k < k 2 1 sin u(z) π t 1 2R(t) 0 dx x 2 1 (tx) 2 k 2 1 k > 1 ( ) where u(z) = z t 1 t (0 1) z U and z is chosen such that k = cosh πr (t) tx 4R(t) R(t) is the Legendre s complete elliptic integral of the first kind and R (t) is complementary integral R(t); p k (z) = 1 + δ k z + in [7] where 8(arccos k) 2 π 2 (1 k 2 ) 0 k < 1 8 (13) δ k = π k = 1 2 π 2 4(k 2 1) k > 1 t(1+t)r 2 (t)

3 A NOTE ON COEFFICIENT INEQUALITIES FOR (j i)-symmetrical FUNCTIONS 79 Definition 13 A function p is said to be in the class k P [A B] 1 B < A 1 k 0 if and only if where p k (z) is defined by (12) and p(z) (A + 1)p k(z) (A 1) (B + 1)p k (z) (B 1) Definition 14 Let i be a positive integer A domain D is said to be i-fold symmetric if a rotation of D about the origin through an angle 2π i carries D onto itself A function f is said to be i-fold symmetric in U if for every z in U f(e 2πi i 2πi z) = e i f(z) The family of all i-fold symmetric functions is denoted by S i and for i = 2 we get class of the odd univalent functions The notion of (j i)-symmetrical functions (i = 2 3 ; j = i 1) is a generalization of the notion of even odd i-symmetrical functions and also generalize the well-known result that each function defined on a symmetrical subset can be uniquely expressed as the sum of an even function and an odd function The theory of (j i) symmetrical functions has many interesting applications for instance in the investigation of the set of fixed points of mappings for the estimation of the absolute value of some integrals and for obtaining some results of the type of Cartan uniqueness theorem for holomorphic mappings [12] Definition 15 Let ε = (e 2πi i ) and j = i 1 where i 2 is a natural number A function f : U C is called (j i)-symmetrical if f(εz) = ε j f(z) z U The family of all (j i)-symmetrical functions is denoted be S (ji) Also S (02) S (12) and S (1i) are called even odd and i-symmetric functions respectively We have the following decomposition theorem Theorem 11 [12] For every mapping f : D C and D is a i-fold symmetric set there exists exactly a unique sequence of (j i)- symmetrical functions f ji i 1 (14) f(z) = f ji (z) then (15) From (14) we have f ji (z) = j=0 ( f ji (z) = 1 i 1 ε vj f(ε v z) = 1 i 1 ) ε vj a n (ε v z) n i i v=0 ψ n a n z n a 1 = 1 n=1 v=0 n=1 ψ n = 1 i 1 ε (n j)v = i v=0 { 1 n = li + j; 0 n li + j;

4 80 AL SARARI AND LATHA where (f A; i = 1 2 ; j = i 1) Definition 16 A function f A is said to be in the class k UC-uniformly convex if and only if R (1 + (zf (z)) ) f k zf (z) (z) 1 f(z) z U k 0 Definition 17 A function f A is said to be in the class k US if and only if ( zf ) (z) R k zf (z) f(z) f(z) 1 z U k 0 Now using the concepts (j i) symmteric points we define the following Definition 18 A function f A is said to be in the class k V (ji) [A B] k 0 1 B < A 1 if and only if (16) R or equivalently zf (B 1) (z) f ji(z) (B + 1) zf (z) f ji(z) (A 1) > k (A + 1) zf (z) k P [A B] f ji (z) (B 1) zf (z) f ji(z) (A 1) (B + 1) zf (z) f ji(z) 1 (A + 1) Definition 19 A function f A is said to be in the class k CV (ji) [A B] k 0 1 B < A 1 if and only if (zf (B 1) (z)) f ji (17) R (z) (A 1) (B 1) (zf (z)) f > k ji (z) (A 1) (B + 1) (zf (z)) f ji (z) (A + 1) (B + 1) (zf (z)) f ji (z) (A + 1) 1 or equivalently (zf (z)) f ji (z) k P [A B] The following special cases are of interest: (i) k V (1i) [A B] = k ST [A B i] k CV (1i) [A B] = k UCV [A B i] the classes introduced by Fuad Alsarari and Latha [4] (ii) k V (11) [A B] = k ST [A B] k CV (11) [A B] = k UCV [A B] the classes introduced by Khalida Inyat Noor and Sarfraz Nawaz Malik [6] (iii) k V (11) [A B] = k Vb σ (A B) the class introduced by Fuad Alsarari and Latha [11] (iv) k V (11) [1 1] = k ST k CV (11) [1 1] = k UCV the well-known classes introduced by Kanas and Wisniowska [8]

5 A NOTE ON COEFFICIENT INEQUALITIES FOR (j i)-symmetrical FUNCTIONS 81 (v) 0 V (ji) [A B] = S (ji) [A B] the class introduced by Fuad Alsarari and S Latha [10] (vi) 0 V (11) [A B] = S [A B] 0 CV (11) [A B] = C[A B] the well-known classes introduced by Janowski [1] We need the following lemmas to prove our main results Lemma 11 [13] Let h(z) = 1 + n=1 c nz n be subordinate to H(z) = 1 + n=1 b nz n If H(z) is univalent in U and H(z) is convex then c n b 1 n 1 Lemma 12 [6] Let h(z) = 1 + n=1 c nz n P [A B] Then c n δ AB δ AB = (A B) δ k 2 where δ k is defined by (13) 2 Main results Theorem 21 A function f A and of the form (11) is in the class k V (ji) [A B] if it satisfies the condition [2(k + 1)(n ψ n ) + n(b + 1) ψ n (A + 1) ] a n (21) < (B + 1) (A + 1)ψ 1 2(k + 1) ψ 1 1 where 1 B < A 1 k 0 and ψ n is defined by (15) Proof Assuming that (22) holds then it suffices to show that (B 1) zf (z) (A 1) zf (z) f k ji (z) (B 1) (A 1) (B + 1) zf 1 (z) (A + 1) R f ji (z) (B + 1) zf 1 < 1 (z) (A + 1) f ji (z) f ji (z) we get k (B 1) zf (z) (A 1) f ji (z) (B + 1) zf (z) f ji (z) 1 (A + 1) R zf (z) (B 1) (A 1) f ji (z) (B + 1) zf (z) f ji (z) 1 (A + 1) (k + 1) (B 1)zf (z) (A 1)f ji (z) (B + 1)zf (z) (A + 1)f ji (z) 1 = 2(k + 1) f ji (z) zf (z) (B + 1)zf (z) (A + 1)f ji (z) = 2(k + 1) (ψ 1 1)z + (ψ n n)a n z n [(B + 1) (A + 1)ψ 1 ]z + [n(b + 1) ψ n(a + 1)]a n z n

6 82 AL SARARI AND LATHA ψ (k + 1) ψ n n a n (B + 1) (A + 1)ψ 1 n(b + 1) ψ n(a + 1) a n The last expression is bounded above by 1 then [2(k + 1) ψ n n + (1 γ) n(b + 1) ψ n (A + 1) ] a n < (B + 1) (A + 1)ψ 1 2(k + 1) ψ 1 1 and this completes the proof Theorem 22 A function f A and of the form (11) is in the class k CV (ji) [A B] if it satisfies the condition n{[2(k+1)(n ψ n )+ n(b+1) ψ n (A+1) ]} a n (22) < (B + 1) (A + 1)ψ 1 2(k + 1) ψ 1 1 where 1 B < A 1 k 0 and ψ n is defined by (15) Particular choices in Theorem 21 and Theorem 22 respectively we get the following corollaries When j = 1 we have the following results proved by Fuad Alsarari and Latha [4] Corollary 21 A function f A and of the form (11) is in the class k ST [A B N] if it satisfies the condition {2(k + 1)(n λ N (n)) + n(b + 1) λ N (n)(a + 1) } a n < B A Corollary 22 A function f A and of the form (11) is in the class k UCV [A B N] if it satisfies the condition n{2(k + 1)(n λ N (n)) + n(b + 1) λ N (n)(a + 1) } a n < B A For j = i = 1 we have the following known results proved by Khalida Inayat Noor and Sarfraz Nawaz Malik [6] Corollary 23 A function f A and form (11) in the class k ST [A B] if it satisfies the condition (23) {2(k + 1)(n 1) + n(b + 1) (A 1) } a n < B A where 1 B < A 1 and k 0

7 A NOTE ON COEFFICIENT INEQUALITIES FOR (j i)-symmetrical FUNCTIONS 83 Corollary 24 A function f A and form (11) in the class k UCV [A B] if it satisfies the condition (24) n{2(k + 1)(n 1) + n(b + 1) (A 1) } a n < B A where 1 B < A 1 and k 0 For j = i = A = B = 1 we have following result due to Kanas and Wisniowska [3] Corollary 25 A function f A and form (11) in the class k ST if it satisfies the condition {n + k(n 1)} a n < 1 k 0 Also for k = 0 j = i = 1 A = 1 2α with 0 α < 1 then we have the following known result Corollary 26 A function f A and form (11) in the class S (α) if it satisfies the condition {(n α)} a n < 1 α where 0 α < 1 Theorem 23 Let f k V (ji) [A B] and is of the form (11) Then for n 2 1 B < A 1 k 0 (25) a n δ k (A B) 2(h ψ h )B where δ k ψ n are defined respectively by (13) (15) where Proof By definition (18) we have then we have we have zf (z) f ji (z) = p(z) p(z) k P [A B] zf (z) = f ji (z) ( 1 + ) c n z n n=1 (1 ψ 1 )z + (n ψ n )z n = ψ n a n z n c n z n n=1 n=1

8 84 AL SARARI AND LATHA Equating coefficients of z n on both sides we have This implies that (n ψ n )a n = ψ n h a n h c h a 1 = ψ 1 = 1 a n By Lemma 12 we get 1 ψ n h a n h c h a 1 = ψ 1 = 1 (n ψ n ) (26) a n δ k (A B) 2(n ψ n ) Now we prove that (27) δ k (A B) 2(n ψ n ) ψ h a h For this we use the induction method For n = 2: from (26) we have From (28) we have For n = 3: from (26) we have From (28) we have a 3 δ k (A B) 2(3 ψ 3 ) ψ h a h a 2 δ k (A B) 2(2 ψ 2 ) a 2 δ k (A B) 2(2 ψ 2 ) δ k (A B) 2(h ψ h )B ) [ 1 + δ ] k (A B) 2(2 ψ 2 ) ψ 2 a 3 δ k (A B) δ k (A B) 2(2 ψ 2 )B 2(2 ψ 2 ) 2(3 ψ 3 ) δ k (A B) δ k (A B) + 2(2 ψ 2 ) 2(2 ψ 2 ) 2(3 ψ 3 ) δ [ k (A B) 1 + δ ] k (A B) 2(3 ψ 3 ) 2(2 ψ 3 ) Let the hypothesis be true for n = m From (26) we have From (28) we have a m δ k (A B) ψ h a h a 1 = ψ 1 = 1 2(m ψ m ) a m δ k (A B) 2(h ψ h )B

9 A NOTE ON COEFFICIENT INEQUALITIES FOR (j i)-symmetrical FUNCTIONS 85 By the induction hypothesis we have δ k (A B) ψ h a h 2(m ψ m ) δ k (A B) + 2(h ψ h ) Multiplying both sides by δ k (A B)+2(m ψ m) 2(m+1 ψ m+1) we have m δ k (A B) + 2(h ψ h ) = δ k (A B) + 2(h ψ h ) δ k (A B) 2(m ψ m ) δ k (A B) + 2(m ψ m ) ψ h a h 2(m + 1 ψ m+1 ) [ δ k (A B) δ k (A B) 2(m + 1 ψ m+1 ) 2(m ψ m ) [ δ k (A B) 2(m + 1 ψ m+1 ) = δ k (A B) 2(m + 1 ψ m+1 ) ψ m a m + δ k (A B) 2(m + 1 ψ m+1 ) m ψ h a h m ψ h a h + m ψ h a h ψ h a h ] ψ h a h ] δ k (A B) + 2(h ψ h ) Which shows that inequality (27) is true for n = m + 1 Hence the required result Theorem 24 Let f k CV (ji) [A B] and is of the form (11) Then for n 2 1 B < A 1 k 0 (28) a n 1 n δ k (A B) 2(h ψ h )B where δ k ψ n are defined respectively by (13) (15) Particular choices in Theorem 23 and Theorem 24 respectively we get the following corollaries When j = 1 we have the following results proved by Fuad Alsarari and Latha [4] Corollary 27 Let f k ST [A B N] and is of the form (11) Then for n 2 a n j=1 δ k (A B) 2(j λ N (j))b 2(j + 1 λ N (j + 1))

10 86 AL SARARI AND LATHA Corollary 28 Let f(z) k UCV [A B N] and is of the form (11) Then for n 2 a n 1 n j=1 δ k (A B) 2(j λ N (j))b 2(j + 1 λ N (j + 1)) When j = i = 1 we have the following results proved Khalida Inayat Noor and Sarfraz Nawaz Malik [6] Corollary 29 Let f k ST [A B] then a n n 2 j=0 δ k (A B) 2jB 1 B < A 1 n 2 2(j + 1) Corollary 210 Let f k UCV [A B] then a n 1 n n 2 j=0 δ k (A B) 2jB 1 B < A 1 n 2 2(j + 1) For A = 1 B = 1 j = i = 1 γ = 0 we arrive at Kanas and Wisniowska [3] Corollary 211 Let f k ST then a n n 2 j=0 δ k + j n 2 (j + 1) Also for k = 0 δ k = 2 j = i = 1 we have the well-known result proved by Janowski [1] Corollary 212 Lat f S [A B] then a n n 2 j=0 (A B) jb 1 B < A 1 n 2 (j + 1) References [1] W Janowski Some extremal problems for certain families of analytic functions Ann Polon Math 28 (1973) [2] A W Goodman Univalent Functions Vols I-II Mariner Publishing Company Florida USA (1983) [3] S Kanas A Wisniowska Conic domains and starlike functions Roumaine Math Pures Appl 45 (2000) [4] F Alsarari and S Latha Conic regions and symmetric points Int J Pure App Math 97(3)(2014) [5] S Latha Coefficient inqualities for certain classes of Ruscheweyh type analytic functions J Inequal Pure Appl Math 9(2)(2008) Art 52 5 pp [6] K L Noor and S N Malik On coefficient inequalities of functions associatd with conic domains Comput Math Appl 62(2011) [7] S Kanas Coefficient estimates in subclasses of the Caratheodory class related to conical domains Acta Math Univ Comenian 74(2)(2005) [8] S Kanas and A Wisniowska Conic regions and k-uniform convexity J comput Appl Math 105(1999)

11 A NOTE ON COEFFICIENT INEQUALITIES FOR (j i)-symmetrical FUNCTIONS 87 [9] H A Al- Kharsani A Sofo Subordination results on harmonic k-uniformly convex mappings and related classes Appl Math Comput Math Appl 59(12)(2010) [10] Fuad Alsarari and S Latha A few results on functions that are Janowski starlike related to (j k)-symmetric opints Octo Math Maga 21(2)(2013) [11] F Alsarari and SLatha A note on cofficent inqualities for certain classes of Ruscheweyh type analytic functions in conic regions accepted for publication in Mathematical Sciences Letters(2016) [12] P Liczberski and J Polubinski On (j k)-symmtrical functions Mathematica Bohemica 120(1995) [13] W Rogosinski On the coefficients of subordiniate function Proc Lond Soc 48(1943) Received by editors ; Revised version on ; Available online Department of Studies in Mathematics Manasagangotri University of Mysore Mysore INDIA address: alsrary@yahoocom Department of Mathematics Yuvaraja s College University of Mysore Mysore INDIA address: drlatha@gmailcom