A study of multivalent q-starlike functions connected with circular domain
Lei Shi, Qaiser Khan, Gautam Srivastava, Jin-Lin Liu, Muhammad Arif
aa r X i v : . [ m a t h . C A ] J u l A STUDY OF MULTIVALENT q -STARLIKE FUNCTIONS CONNECTEDWITH CIRCULAR DOMAIN LEI SHI , QAISER KHAN , GAUTAM SRIVASTAVA , , JIN-LIN LIU , MUHAMMAD ARIF , ∗ Abstract.
In the present article, our aim is to examine some useful problems includingthe convolution problem, sufficiency criteria, coefficient estimates and Fekete-Szeg¨o typeinequalities for a new subfamily of analytic and multivalent functions associated with circulardomain. In addition, we also define and study a Bernardi integral operator in its q -extensionfor multivalent functions. Introduction
The study of the q -extension of calculus or the q -analysis attracted and motivated many re-searchers becauuse of its applications in different parts of mathematical sciences. Jackson[15, 14]) was one of the main contributer among all the mathematicians who initiated and es-tablished the theory of q -calculus. As an interesting sequel to [13], in which use was made of the q -derivative operator for the first time for studying the geometry of q -starlike functions, a firmfooting of the usage of the q -calculus in the context of Geometric Function Theory was actuallyprovided and the basic (or q -) hypergeometric functions were first used in Geometric FunctionTheory in a book chapter by Srivastava (see, for details, [29, pp. 347 et seq. ]). The theory of q -starlike functions was later extended to various families of q -starlike functions by (for exam-ple) Agrawal and Sahoo [1] (see also the recent investigations on this subject by Srivastava etal. [32, 33, 34, 35, 36, 37]). Motivated by these q -developments in Geometric Function Theory,many authors such as like Srivastava and Bansal [29] were added their contributions in thisdirection which has made this research area much more attractive.In 2014, Kanas and R˘aducanu [17] used the familiar Hadamad product to define a q -extensionof the Ruscheweyh operator and discussed important applications of this operator. Moreover,the extensive study of this q -Ruscheweyh operator was further made by Mohammad and Darus[5] and Mahmood and Sok´o l [20]. Recently, a new idea was presented by Darus [21] andintroduced a new differential operator called generalized q -differential operator with the helpof q -hypergeometric functions where they studied some useful applications of this operator.For the recent extension of different operators in q -analogue, see the references [2, 9, 8]. Theoperator defined in [17] was extended further for multivalent functions by Arif et al. [10] inwhich they investigated its important applications. The aim of this paper is to define a familyof multivalent q -starlike functions associated with circular domain and to study some of itsuseful properties.Let A p ( p ∈ N = { , , , . . . } ) contains all multivalent functions say f that are holomorphic oranalytic in a subset D = { z : | z | < } of a complex plane C and having the series form: f ( z ) = z p + ∞ X l =1 a l + p z l + p , ( z ∈ D ) . (1.1)For two analytic functions f and g in D , then f is subordinate to g, symbolically presented as f ≺ g or f ( z ) ≺ g ( z ) , if we can find an analytic function w with the properties w (0) = 0 & Date : 16 May 2019 ∗ Corresponding author2010
Mathematics Subject Classification.
Key words and phrases.
Analytic functions, Multivalent functions, Janowski functions, Differential subordi-nations, q -derivative, q -Starlike functions, Fekete-Szeg¨o type inequalities. | w ( z ) | < f ( z ) = g ( w ( z )) ( z ∈ D ) . Also, if g is univalent in D , then we have: f ( z ) ≺ g ( z ) ( z ∈ D ) ⇐⇒ f (0) = g (0) and f ( D ) ⊂ g ( D ) . For given q ∈ (0 , q -analogue of f is given by ∂ q f ( z ) = f ( z ) − f ( qz ) z (1 − q ) , ( z = 0 , q = 1) . (1.2)Making (1 .
1) and (1 . , we easily get that for n ∈ N and z ∈ D ∂ q ( ∞ X n =1 a n + p z n + p ) = ∞ X n =1 [ n + p, q ] a n + p z n + p − , (1.3)where [ n, q ] = 1 − q n − q = 1 + n − X l =1 q l , [0 , q ] = 0 . For n ∈ Z ∗ := Z \ {− , − , . . . } , the q -number shift factorial is given as:[ n, q ]! = (cid:26) , n = 0 , [1 , q ] [2 , q ] . . . [ n, q ] , n ∈ N . Also, with x >
0, the q -analogue of the Pochhammer symbol has the form:[ x, q ] n = (cid:26) , n = 0 , [ x, q ][ x + 1 , q ] · · · [ x + n − , q ] , n ∈ N , and, for x >
0, the Gamma function in q -analogue is presented asΓ q ( x + 1) = [ x, q ] Γ q ( t ) and Γ q (1) = 1 . We now consider a functionΦ p ( q, µ + 1; z ) = z p + ∞ X n =2 Λ n + p z n + p , ( µ > − , z ∈ D ) , (1.4)with Λ n + p = [ µ + 1 , q ] n + p [ n + p, q ]! . (1.5)The series defined in (1 .
4) is converges absolutely in D . Using Φ p ( q, µ ; z ) with µ > − L µ + p − q : A p → A p by L µ + p − q f ( z ) = Φ p ( q, µ ; z ) ∗ f ( z ) = z p + ∞ X n =2 Λ n + p a n + p z n + p , ( z ∈ D ) , (1.6)Also we note thatlim q → − Φ p ( q, µ ; z ) = z p (1 − z ) µ +1 and lim q → − L µ + p − q f ( z ) = f ( z ) ∗ z p (1 − z ) µ +1 . Now when q → − , the operator defined in (1 .
6) becomes to the familiar differential operatorinvestigated in [12] and further, setting p = 1 , we get the most familiar operator known asRuscheweyh operator [24] (see also [3, 22]). Also, for different types of operators in q -analogue,see the works [2, 4, 6, 7, 9, 21].Motivated from the work studied in [8, 13, 19, 25, 31], we establish a family S ∗ p ( q, µ, A, B ) usingthe operator L µ + p − q as: Definition 1.1.
Suppose that q ∈ (0 ,
1) and − ≦ B < A ≦ . Then f ∈ A p belongs to the set S ∗ p ( q, µ, A, B ) , if it satisfies z∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) ≺ Az Bz , (1.7)where the function Az Bz is known as Janowski function studied in [16]. STUDY OF MULTIVALENT q -STARLIKE FUNCTIONS 3 Alternatively, f ∈ S ∗ p ( q, µ, A, B ) ⇔ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z∂ q L µ + p − q f ( z )[ p,q ] L µ + p − q f ( z ) − A − B z∂ q L µ + p − q f ( z )[ p,q ] L µ + p − q f ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < . (1.8)2. A Set of Lemmas
Lemma 2.1. [23]
Let h ( z ) = 1 + ∞ P n =1 d n z n ≺ K ( z ) = 1 + ∞ P n =1 k n z n in D . If K ( z ) is convexunivalent in D , then | d n | ≦ | k | , for n ≧ . Lemma 2.2.
Let W contain all functions w that are analytic in D which satisfying w (0) = 0 & | w ( z ) | < . If the function w ∈ W given by w ( z ) = ∞ X k =1 w k z k ( z ∈ D ) . then for λ ∈ C , we have (cid:12)(cid:12) w − λw (cid:12)(cid:12) ≦ max { | λ |} , (2.1) and (cid:12)(cid:12)(cid:12)(cid:12) w + 14 w w + 116 w (cid:12)(cid:12)(cid:12)(cid:12) ≦ . (2.2) These results are best possible.
For the first and second part see the reference [18] and [28] respectively.3.
The Main Results and Their Consequences
Theorem 3.1.
Let f ∈ A p has the series form (1 . and satisfing the inequality given by ∞ X n =1 ∧ n + p ([ n + p, q ] (1 − B ) − [ p, q ] (1 − A )) | a n + p | ≦ [ p, q ] ( A − B ) . (3.1) Then f ∈ S ∗ p ( q, µ, A, B ) . Proof.
To show f ∈ S ∗ p ( q, µ, A, B ) , we just need to show the relation (1 . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z∂ q L µ + p − q f ( z )[ p,q ] L µ + p − q f ( z ) − A − B z∂ q L µ + p − q f ( z )[ p,q ] L µ + p − q f ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z∂ q L µ + p − q f ( z ) − [ p, q ] L µ + p − q f ( z ) A [ p, q ] L µ + p − q f ( z ) − Bz∂ q L µ + p − q f ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Using (1 . .
1) and (1 . (cid:12)(cid:12)(cid:12)(cid:12) [ p,q ] z p + P ∞ n =1 ∧ n + p a n + p [ n + p,q ] z n + p − [ p,q ] ( z p + P ∞ n =1 ∧ n + p a n + p z n + p ) A [ p,q ] ( z p + P ∞ n =1 ∧ n + p a n + p z n + p ) − B ( [ p,q ] z p + P ∞ n =1 ∧ n + p a n + p [ n + p,q ] z n + p ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) P ∞ n =1 ∧ n + p a n + P ([ n + p,q ] − [ p,q ]) z n + p ( A − B )[ p,q ] z p + P ∞ n =1 ∧ n + p a n + p ( A [ p,q ] − B [ n + p,q ]) z n + p (cid:12)(cid:12)(cid:12) ≦ P ∞ n =1 ∧ n + p | a n + P | ([ n + p,q ] − [ p,q ]) | z | n + p ( A − B )[ p,q ] | z | p − P ∞ n =1 ∧ n + p | a n + p | ( A [ p,q ] − B [ n + p,q ]) | z | n + p ≦ P ∞ n =1 ∧ n + p | a n + P | ([ n + p,q ] − [ p,q ])( A − B )[ p,q ] − P ∞ n =1 ∧ n + p | a n + p | ( A [ p,q ] − B [ n + p,q ]) < , where we have used the inequality (3 .
1) and this completes the proof. (cid:3)
Varying the parameters µ, b, A and B in the last Theorem, we get the following known resultsdiscussed earlier in [26]. L. SHI, Q. KHAN, G. SRIVASTAVA, J.-L. LIU, M. ARIF
Corollary 3.2.
Let f ∈ A be given by (1 . and satisfy the inequality ∞ X n =2 ([ n, q ] (1 − B ) − A ) | a n | ≦ A − B. Then the function f ∈ S ∗ q [ A, B ] . By choosing q → − in the last corollary, we get the known result proved by Ahuja [3] andfurthermore for A = 1 − α and B = − , we obtain the result for the family S ∗ ( ξ ) which wasproved by Silverman [27]. Theorem 3.3.
Let f ∈ S ∗ p ( q, µ, A, B ) be of the form (1 . . Then | a p +1 | ≦ ψ ( A − B ) ∧ p , (3.2) and for n ≧ , | a n + p | ≦ ( A − B ) ψ n ∧ n + p n − Y t =1 (cid:18) p, q ] ( A − B )([ p + t, q ] − [ p, q ]) (cid:19) , (3.3) where ψ n := ψ n ( p, q ) = [ p, q ]([ n + p, q ] − [ p, q ]) . (3.4) Proof. If f ∈ S ∗ p ( q, µ, A, B ) , then by definition we have z∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) = 1 + Aw ( z )1 + Bw ( z ) . (3.5)Let us put p ( z ) = 1 + ∞ X n =1 d n z n = 1 + Aw ( z )1 + Bw ( z ) . Then by Lemma 2.1, we get | d n | ≦ A − B. (3.6)Now, from (3 .
5) and (1 . z p + ∞ P n =1 [ n + p,q ][ p,q ] Λ n + p a n + p z n + p = (cid:18) ∞ P n =1 d n z n (cid:19) (cid:18) z p + ∞ P n =1 Λ n + p a n + p z n + p (cid:19) . (3.7)Equating coefficients of z n + p on both sides ∧ n + p ([ n + p, q ] − [ p, q ]) a n + p = [ p, q ] ∧ n + p − a n + p − d + · · · + [ p, q ] ∧ p a p d n − . Taking absolute on both sides and then using (3 . , we have ∧ n + p ([ n + p, q ] − [ p, q ]) | a n + p | ≦ [ p, q ] ( A − B ) n − X k =1 ∧ k + p | a k + p | ! , and this further implies | a n + p | ≦ ( A − B ) ψ n ∧ n + p n − X k =1 ∧ k + p | a k + p | ! , (3.8)where ψ n is given by (3 . . So for n = 1, we have from (3 . | a p +1 | ≦ ( A − B ) ψ ∧ p , and this shows that (3 .
2) holds for n = 1 . To prove (3 .
3) we apply mathematical induction.Therefore for n = 2, we have from (3 . | a p +2 | ≦ ( A − B ) ψ ∧ p (1 + ∧ p | a p | ) , STUDY OF MULTIVALENT q -STARLIKE FUNCTIONS 5 using (3 . | a p +2 | ≦ ( A − B ) ψ ∧ p (1 + ( A − B ) ψ ) , which clearly shows that (3 .
3) holds for n = 2. Let us assume that (3 .
3) is true for n ≦ m − , that is, | a m − p | ≦ ( A − B ) ψ m − ∧ m + p − m − Y t =1 (1 + ( A − B ) ψ t ) . Consider | a m + p | ≦ ( A − B ) ψ m ∧ m + p m − X k =1 ∧ k + p | a k + p | ! = ( A − B ) ψ m ∧ m + p ( A − B ) ψ + . . . + ( A − B ) ψ m − m − Y t =1 (1 + ( A − B ) ψ t ) ) = ( A − B ) ψ m ∧ m + p m − Y t =1 (cid:18) p, q ] ( A − B )([ p + t, q ] − [ p, q ]) (cid:19) , and this implies that the given result is true for n = m. Hence, using mathematical induction,we achived the inequality (3 . . (cid:3) Theorem 3.4.
Let f ∈ S ∗ p ( q, µ, A, B ) and be given by (1 . . Then for λ ∈ C (cid:12)(cid:12) a p +2 − λa p +1 (cid:12)(cid:12) ≦ ( A − B ) ψ Λ p +2 { | υ |} , where υ is given by υ = ( B − ( A − B ) ψ ) + Λ p +2 ψ Λ p +1 ψ ( A − B ) λ. (3.9) Proof.
Let f ∈ S ∗ p ( q, µ, A, B ) and consider the right hand side of (3 .
5) we have1 + Aw ( z )1 + Bw ( z ) = A ∞ X k =1 w k z k ! B ∞ X k =1 w k z k ! − , where w ( z ) = ∞ X k =1 w k z k , and after simple computations, we can rewrite1 + Aw ( z )1 + Bw ( z ) = 1 + ( A − B ) w z + ( A − B ) (cid:8) w − Bw (cid:9) z + . . . . (3.10)Now, left hand side of (3 . z∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) = ∞ X n =1 [ n + p, q ][ p, q ] Λ n + p a n + p z n ! ∞ X n =1 Λ n + p a n + p z n ! − = 1 + Λ p ψ a p z + Λ p a p ψ − Λ p a p ψ ! z + . . . . (3.11)From (3 .
10) and (3 . , we have a p +1 = ψ Λ p +1 ( A − B ) w a p +2 = ( A − B ) ψ Λ p +2 (cid:8) w + (( A − B ) ψ − B ) w (cid:9) . L. SHI, Q. KHAN, G. SRIVASTAVA, J.-L. LIU, M. ARIF
Now consider (cid:12)(cid:12) a p +2 − λa p +1 (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( A − B ) ψ Λ p +2 (cid:8) w + (( A − B ) ψ − B ) w (cid:9) − λ ψ Λ p +1 ( A − B ) w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( A − B ) ψ Λ p +2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w − ( ( B − ( A − B ) ψ ) + Λ p +2 ψ Λ p +1 ψ ( A − B ) λ ) w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , using Lemma 2.2, we have (cid:12)(cid:12) a p +2 − λa p +1 (cid:12)(cid:12) ≦ ( A − B ) ψ Λ p +2 { | υ |} , where υ is given by υ = ( B − ( A − B ) ψ ) + Λ p +2 ψ Λ p +1 ψ ( A − B ) λ. This completes the proof. (cid:3)
Theorem 3.5.
Let f ∈ S ∗ p ( q, µ, A, B ) and be given by (1 . . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a p +3 − q + 2 q + q + 1 Λ p Λ p Λ p a p +2 a p +1 + 1[3 , q ] Λ p Λ p a p +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≦ ( A − B ) ( B − + 18Λ p ) ψ , where ψ n and ∧ n + p are defined by (3 . and (1 . respectively.Proof. From the relations (3 .
10) and (3 . , we have a p +3 − q + 2 q + q + 1 Λ p Λ p Λ p a p +2 a p +1 + 1[3 , q ] Λ p Λ p a p +1 ! = ( A − B ) ψ Λ p (cid:8) w − Bw w + B w (cid:9) , equivalently, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a p +3 − q + 2 q + q + 1 Λ p Λ p Λ p a p +2 a p +1 + 1[3 , q ] Λ p Λ p a p +1 !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( A − B ) ψ Λ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) w + 14 w w + 116 w (cid:19) − B − (cid:0) w − w (cid:1) + 16 B − B − w (cid:12)(cid:12)(cid:12)(cid:12) ≦ ( A − B ) ψ Λ p (cid:26) B −
116 + 16 B − B − (cid:27) ≦ ( A − B ) ψ Λ p (cid:26) B − B + 58 (cid:27) , where we have used (2 .
1) and (2 . . This completes the proof. (cid:3)
Theorem 3.6.
Let f ∈ A p be given by (1 . . Then the function f is in the class S ∗ p ( q, µ, A, B ) , if and only if e iθ ( B − [ p, q ] A ) z (cid:20) L µ + p − q f ( z ) ∗ (cid:18) ( N + 1) z p − qLz p +1 (1 − z ) (1 − qz ) (cid:19)(cid:21) = 0 , (3.12) for all N = N θ = ([ p, q ] − e − iθ ([ p, q ] A − B ) , (3.13) L = L θ = (cid:0) e − iθ + [ p, q ] A (cid:1) ([ p, q ] A − B ) , and also for N = 0 , L = 1 . STUDY OF MULTIVALENT q -STARLIKE FUNCTIONS 7 Proof.
Since the function f ∈ S ∗ p ( q, µ, A, B ) is analytic in D , it implies that L µ + p − q f ( z ) = 0for all z ∈ D ∗ = D \{ } ; that is e iθ ( B − [ p, q ] A ) z L µ + p − q f ( z ) = 0 ( z ∈ D ) , and this is equivalent to (3 .
12) for N = 0 and L = 1 . From (1 . , according to the definitionof the subordination, there exists an analytic function w with the property that w (0) = 0 and | w ( z ) | < z∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) = 1 + Aω ( z )1 + Bω ( z ) ( z ∈ D ) , which is equalent for z ∈ D , ≦ θ < πz∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) = 1 + Ae iθ Be iθ , (3.14)and further written in more simplified form (cid:0) Be iθ (cid:1) z∂ q L µ + p − q f ( z ) − [ p, q ] (cid:0) Ae iθ (cid:1) L µ + p − q f ( z ) = 0 . (3.15)Now using the following convolution properties in (3 . L µ + p − q f ( z ) ∗ z p (1 − z ) = L µ + p − q f ( z ) and L µ + p − q f ( z ) ∗ z p (1 − z )(1 − qz ) = z∂ q L µ + p − q f ( z ) , and then simple computation gives1 z " L µ + p − q f ( z ) ∗ (cid:0) Be iθ (cid:1) z p (1 − z ) (1 − qz ) − [ p, q ] (cid:0) Ae iθ (cid:1) z p (1 − z ) ! = 0 , or equivalently ( B − [ p, q ] A ) e iθ z (cid:20) L µ + p − q f ( z ) ∗ (cid:18) ( N + 1) z p − Lqz p +1 (1 − z ) (1 − qz ) (cid:19)(cid:21) = 0 , which is the required direct part.Assume that (3 .
1) holds true for L θ − N θ = 0, it follows that e iθ ( B − [ p, q ] A ) z L µ + p − q f ( z ) = 0 , for all z ∈ D . Thus the function h ( z ) = z∂ q L µ + p − q f ( z )[ p,q ] L µ + p − q f ( z ) is analytic in D and h (0) = 1 . Since we have shownthat (3 .
15) and (3 .
1) are equivalent, therefore we have z∂ q L µ + p − q f ( z )[ p, q ] L µ + p − q f ( z ) = 1 + Ae iθ Be iθ ( z ∈ D ) . (3.16)Suppose that H ( z ) = 1 + Az Bz , z ∈ D . Now from relation (3 .
16) it is clear that H ( ∂ D ) ∩ h ( D ) = φ. Therefore the simply connecteddomain h ( D ) is contained in a connected component of C \ H ( ∂ D ) . The univalence of thefunction h together with the fact H (0) = h (0) = 1 shows that h ≺ H which shows that f ∈ S ∗ p ( q, µ, A, B ) . (cid:3) We now define an integral operator for the function f ∈ A p as follows; L. SHI, Q. KHAN, G. SRIVASTAVA, J.-L. LIU, M. ARIF
Definition 3.7.
Let f ∈ A p . Then L : A p → A p is called the q -analogue of Benardi integraloperator for multivalent functions defined by L ( f ) = F η,p with η > − p, where F η,p is given by F η,p ( z ) = [ η + p, q ] z η z Z t η − f ( t ) d q t, (3.17)= z p + ∞ X n =1 [ η + p, q ][ η + p + n, q ] a n + p z n + p , ( z ∈ D ) . (3.18)We easily obtain that the series defined in (3 .
18) is converges absolutely in D . Now If q → F η,p reduces to the integral operator studied in [31] and further by taking p = 1 , we obtain the familiar Bernardi integral operator introduced in [11]. Theorem 3.8. If f is of the form (1 . belongs to the family S ∗ p ( q, µ, A, B ) and F η,p ( z ) = z p + ∞ X n =1 b n + p z n + p , (3.19) where F η,p is the integral operator given by (3 . , then | b p +1 | ≦ [ η + p, q ][ η + p + 1 , q ] ψ ( A − B ) ∧ p and for n ≧ | b p + n | ≦ [ η + p, q ][ η + p + n, q ] ( A − B ) ψ n ∧ n + p n − Y t =1 (cid:18) p, q ] ( A − B )([ p + t, q ] − [ p, q ]) (cid:19) , where ψ n and ∧ n + p are defined by (3 . and (1 . respectively.Proof. The proof follows easily by using (3 .
18) and Theorem 3.3. (cid:3)
Theorem 3.9.
Let f ∈ S ∗ p ( q, µ, A, B ) and be given by (1 . . Also if F η,p is the integral operatordefined by (3 . and is of the form (3 . , then for σ ∈ C (cid:12)(cid:12) b p +2 − σb p +1 (cid:12)(cid:12) ≦ [ η + p, q ][ η + p + 2 , q ] ( A − B ) ψ Λ p +2 { | υ |} , where υ = ( B − ( A − B ) ψ ) + Λ p +2 ψ Λ p +1 ψ ( A − B ) [ η + p, q ][ η + p + 2 , q ][ η + p + 1 , q ] σ. (3.20) Proof.
From (3 .
18) and (3 . , we easily have b p +1 = [ η + p, q ][ η + p + 1 , q ] a p +1 ,b p +2 = [ η + p, q ][ η + p + 2 , q ] a p +2 . Now (cid:12)(cid:12) b p +2 − σb p +1 (cid:12)(cid:12) = [ η + p, q ][ η + p + 2 , q ] (cid:12)(cid:12)(cid:12)(cid:12) a p +2 − σ [ η + p, q ][ η + p + 2 , q ][ η + p + 1 , q ] a p +1 (cid:12)(cid:12)(cid:12)(cid:12) , ≦ [ η + p, q ][ η + p + 2 , q ] ( A − B ) ψ Λ p +2 { | υ |} , where υ is given by (3 .
20) and we have used Theorem 3.4 to complete the proof. (cid:3)
Conflicts of Interest
The authors agree with the contents of the manuscript and there are no conflicts of interestamong the authors.
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