Alied Mathematics and Comutation 7 (2007) 35 46 www.elsevier.com/locate/amc Coefficient bounds for -valent functions Rosihan M. Ali *, V. Ravichandran, N. Seenivasagan School of Mathematical Sciences, Universiti Sains Malaysia, 00 USM Penang, Malaysia Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday Abstract Shar bounds for ja 2 la 2 j and ja +3j are derived for certain -valent analytic functions. These are alied to obtain Fekete-Szegö like inequalities for several classes of functions defined by convolution. Ó 2006 Elsevier Inc. All rights reserved. Keywords Analytic functions; Starlike functions; Convex functions; -Valent functions; Subordination; Convolution; Fekete-Szegö inequalities. Introduction Let A be the class of all functions of the form f ðzþ ¼z X n¼ a n z n ðþ which are analytic in the oen unit disk D ¼ {zjzj < } and let A ¼ A. For f(z) given by () and g(z) given by gðzþ ¼z P n¼ b nz n, their convolution (or Hadamard roduct), denoted by f * g, is defined by ðf gþðzþ ¼ z X a n b n z n n¼ The function f(z) is subordinate to the function g(z), written f(z) g(z), rovided there is an analytic function w(z) defined on D with w(0) = 0 and jw(z)j < such that f(z)=g(w(z)). Let u be an analytic function with ositive real art in the unit disk D with u(0) = and u 0 (0) > 0 that mas D onto a region starlike with resect to and symmetric with resect to the real axis. We define the class S b; ðuþ to be the subclass of A consisting of functions f(z) satisfying * Corresonding author. E-mail addresses rosihan@cs.usm.my (R.M. Ali), vravi@cs.usm.my (V. Ravichandran), vasagan2000@yahoo.co.in (N. Seenivasagan). 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi0.06/j.amc.2006.0.00
36 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 zf 0 ðzþ b f ðzþ uðzþ; ðz 2 D and b 2 C nf0gþ As secial cases, let S ðuþ ¼ S ; ðuþ; S b ðuþ ¼ S b; ðuþ; S ðuþ ¼ S ; ðuþ For a fixed analytic function g 2 A with ositive coefficients, define the class S b;;gðuþ to be the class of all functions f 2 A satisfying f g 2 S b;ðuþ. This class includes as secial cases several other classes studied in the literature. For examle, when gðzþ ¼z P n¼ n zn, the class S b;;g ðuþ reduces to the class C b,(u) consisting of functions f 2 A satisfying b b zf 00 ðzþ uðzþ; f 0 ðzþ ðz 2 D and b 2 C nf0gþ The classes S * (u) andc(u) ¼ C, (u) were introduced and studied by Ma and Minda [9]. Define the class R b, (u) to be the class of all functions f 2 A satisfying f 0 ðzþ uðzþ; ðz 2 D and b 2 C nf0gþ; b z and for a fixed function g with ositive coefficients, let R b,,g (u) be the class of all functions f 2 A satisfying f * g 2 R b, (u). Several authors [4,,0,6,7,2] have studied the classes of analytic functions defined by using the exression zf 0 ðzþ f 00 ðzþ. We shall also consider a class defined by the corresonding quantity for -valent functions. f ðzþ f ðzþ Define the class S ða; uþ to be the class of all functions f 2 A satisfying að Þ zf 0 ðzþ f ðzþ a z 2 f 00 ðzþ uðzþ f ðzþ ðz 2 D and a P 0Þ Note that S ð0; uþ is the class S ðuþ. Let S ;g ða; uþ be the class of all functions f 2 A for which f g 2 S ða; uþ. We shall also consider the class L M ða; uþ consisting of -valent a-convex functions with resect to u. These are functions f 2 A satisfying a zf 0 ðzþ f ðzþ a zf 00 ðzþ uðzþ f 0 ðzþ ðz 2 D and a P 0Þ Further let M (a,u) be the class of functions f 2 A satisfying zf 0 a ðzþ zf 00 ðzþ f ðzþ f 0 ðzþ a uðzþ ðz 2 D and a P 0Þ Functions in this class are called logarithmic -valent a-convex functions with resect to u. In this aer, we obtain Fekete-Szegö inequalities and bounds for the coefficient a +3 for the classes S ðuþ and S ;gðuþ. These results are then extended to the other classes defined earlier. See [ 7,9,3,4,] for Fekete- Szegö roblem for certain related classes of functions. Let X be the class of analytic functions of the form wðzþ ¼w z w 2 z 2 ð2þ in the unit disk D satisfying the condition jw(z)j <. We need the following lemmas to rove our main results. Lemma. If w 2 X, then >< t if t 6 ; jw 2 tw 2 j 6 if 6 t 6 ; > t if t P When t < ort>, equality holds if and only if w(z) = z or one of its rotations. If < t <, then equality holds if and only if w(z) = z 2 or one of its rotations. Equality holds for t = if and only if wðzþ ¼
z kz kz R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 37 ð0 6 k 6 Þ or one of its rotations while for t =, equality holds if and only if wðzþ ¼ z kz kz or one of its rotations. Also the shar uer bound above can be imroved as follows when < t < ð0 6 k 6 Þ jw 2 tw 2 jðt Þjw j 2 6 ð < t 6 0Þ and jw 2 tw 2 jð tþjw j 2 6 ð0 < t < Þ Lemma is a reformulation of a Lemma of Ma and Minda [9]. Lemma 2 [5, Inequality 7,. 0]. If w 2 X, then, for any comlex number t, jw 2 tw 2 j 6 maxf; jtjg The result is shar for the functions w(z) = z 2 or w(z) = z. Lemma 3 []. If w 2 X, then for any real numbers q and q 2, the following shar estimate holds jw 3 q w w 2 q 2 w 3 j 6 Hðq ; q 2 Þ; ð3þ where for ðq ; q 2 Þ2D [ D 2 ; jq 2 j for ðq ; q 2 Þ2 S7 >< 2 Hðq ; q 2 Þ¼ ðjq 2 3 jþ for ðq ; q 2 Þ2D [ D 9 ; 3 q 2 q 2 4 q 2 4q 2 jq j 3ðjq jq 2 Þ k¼3 D k ; q 2 2 for ðq ; q 2 Þ2D 0 [ D f2; g; 4 3ðq 2 Þ > 2 ðjq 2 3 j Þ for ðq ; q 2 Þ2D 2 jq j 3ðjq j q 2 Þ The extremal functions, u to rotations, are of the form wðzþ ¼z 3 ; wðzþ ¼z; wðzþ ¼w 0 ðzþ ¼ zð½ð kþe 2 ke Š e e 2 zþ ; ½ð kþe ke 2 Šz wðzþ ¼w ðzþ ¼ zðt zþ t z ; wðzþ ¼w 2ðzÞ ¼ zðt 2 zþ t 2 z ; je j¼je 2 j¼; e ¼ t 0 e ih 0 2 ða bþ; e2 ¼ e ih 0 2 ðia bþ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ t 0 cos h 0 2 ; b ¼ h t 2 0 0 sin2 ; k ¼ b a 2 2b ; t 0 ¼ 2q 2ðq 2 2Þ 3q2 2 3ðq 2 Þðq 2 4q ; t ¼ 2Þ t 2 ¼ jq j 2; 3ðjq jq 2 Þ jq j 2; h 0 cos 3ðjq j q 2 Þ 2 ¼ q q 2 ðq 2 Þ 2ðq2 2Þ 2 2q 2 ðq 2 2Þ 3q2
3 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 The sets D k,k=,2,...,2, are defined as follows D ¼ ðq ; q 2 Þ jq j 6 2 ; jq 2j 6 ; D 2 ¼ ðq ; q 2 Þ 2 6 jq j 6 2; 4 27 ðjq jþ 3 ðjq jþ 6 q 2 6 ; D 3 ¼ ðq ; q 2 Þ jq j 6 2 ; q 2 6 ; D 4 ¼ ðq ; q 2 Þ jq j P 2 ; q 2 6 2 3 ðjq jþ ; D 5 ¼fðq ; q 2 Þ jq j 6 2; q 2 P g; D 6 ¼ ðq ; q 2 Þ 2 6 jq j 6 4; q 2 P 2 ðq2 Þ ; D 7 ¼ ðq ; q 2 Þ jq j P 4; q 2 P 2 3 ðjq j Þ ; D ¼ ðq ; q 2 Þ 2 6 jq j 6 2; 2 3 ðjq jþ 6 q 2 6 4 27 ðjq jþ 3 ðjq jþ ; D 9 ¼ ðq ; q 2 Þ jq j P 2; 2 3 ðjq jþ 6 q 2 6 2jq jðjq jþ q 2 2jq ; j4 D 0 ¼ ðq ; q 2 Þ 2 6 jq j 6 4; 2jq jðjq jþ q 2 2jq j4 6 q 2 6 2 ðq2 Þ ; D ¼ ðq ; q 2 Þ jq j P 4; 2jq jðjq jþ q 2 2jq j4 6 q 2 6 2jq jðjq j Þ q 2 2jq ; j4 D 2 ¼ ðq ; q 2 Þ jq j P 4; 2jq jðjq j Þ q 2 2jq j4 6 q 2 6 2 3 ðjq j Þ 2. Coefficient bounds By making use of the Lemmas 3, we rove the following bounds for the class S ;g ðuþ Theorem. Let u(z) = + B z+b 2 z 2 +B 3 z 3 +, and r ¼ B 2 B B 2 ; r 2B 2 2 ¼ B 2 B B 2 ; r 2B 2 3 ¼ B 2 B 2 2B 2 If f(z) given by () belongs to S ðuþ, then B >< 2 2 ð 2lÞB 2 if l 6 r ; a 2 la 2 B 6 if r 2 6 l 6 r 2 ; > B 2 2 ð 2lÞB 2 if l P r 2 if r 6 l 6 r 3, then a 2 la 2 B 2 ð2l ÞB 2B B If r 3 6 l 6 r 2, then a 2 la 2 B 2 ð2l ÞB 2B B ja j 2 6 B 2 ð5þ ja j 2 6 B 2 ð6þ ð4þ
R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 39 For any comlex number l, a 2 la 2 6 B 2 max ; B 2 ð 2lÞB B ð7þ ja 3 j 6 B 3 Hðq ; q 2 Þ; ðþ where H(q,q 2 ) is as defined in Lemma 3, q ¼ 4B 2 3B 2 and q 2B 2 ¼ 2B 3 3B B 2 2 B 3 2B These results are shar. Proof. If f ðzþ 2S ðuþ; then there is an analytic function w(z) =w z + w 2 z 2 + 2X such that zf 0 ðzþ f ðzþ ¼ uðwðzþþ Since zf 0 ðzþ f ðzþ ¼ a z a2 2a 2 we have from (9), a ¼ B w ; a 2 ¼ 2 B w 2 ðb 2 B 2 Þw2! z 2 3a 3 3 a a 2 a3! z 3 ; ð9þ ð0þ ðþ and a 3 ¼ B 3 w 3 4B 2 3B 2 w w 2 2B 3 3B B 2 2 B 3 w 3 2B 2B ð2þ Using (0) and (), we have a 2 la 2 ¼ B w 2 vw 2 ; ð3þ 2 where v ¼ B ð2l Þ B 2 B The results (4) (6) are established by an alication of Lemma, inequality (7) by Lemma 2 and () follows from Lemma 3. To show that the bounds in (4) (6) are shar, we define the functions K un (n =2,3,...)by zk 0 un ðzþ K un ðzþ ¼ uðzn Þ; K un ð0þ ¼0 ¼½K un Š 0 ð0þ and the functions F k and G k (0 6 k 6 ) by zf 0 k ðzþ zðz kþ ¼ u ; F k ð0þ ¼0 ¼ F 0 k F k ðzþ kz ð0þ and zg 0 k ðzþ zðz kþ ¼ u ; G k ð0þ ¼0 ¼ G 0 k G k ðzþ kz ð0þ Clearly the functions K un ; F k ; G k 2 S ðuþ. We shall also write K u ¼ K u2.
40 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 If l < r or l > r 2, then equality holds if and only if f is K u or one of its rotations. When r < l < r 2, then equality holds if and only if f is K u3 or one of its rotations. If l = r then equality holds if and only if f is F k or one of its rotations. Equality holds for l = r 2 if and only if f is G k or one of its rotations. h Corollary. Let u(z)=+b z+b 2 z 2 +B 3 z 3 +, and let r ¼ g2 B 2 B B 2 ; r g 2 2B 2 2 ¼ g2 B 2 B B 2 ; r g 2 2B 2 3 ¼ g2 B 2 B 2 g 2 2B 2 If f(z) given by () belongs to S ;gðuþ, then 2g 2 B 2 2 g 2 l B 2 g 2 if l 6 r ; >< a 2 la 2 B 6 2g 2 if r 6 l 6 r 2 ; 2g 2 B 2 2 g 2 > B 2 if l P r 2 g 2 l if r 6 l 6 r 3, then a 2 la 2 g2 B 2 2g 2 B B If r 3 6 l 6 r 2, then a 2 la 2 g2 2g 2 B B 2 B 2 g 2 l g 2 2 g 2 l g 2!B!!B! ja j 2 6 B 2g 2 ja j 2 6 B 2g 2 For any comlex number l, ( a 2 la 2 6 B max ; B 2 2 g! ) 2 l B 2g 2 B g 2 ja 3 j 6 B 3g 3 Hðq ; q 2 Þ; ð4þ where H(q,q 2 ) is as defined in Lemma 3, q ¼ 4B 2 3B 2 and q 2B 2 ¼ 2B 3 3B B 2 2 B 3 2B These results are shar. Theorem 2. Let u be as in Theorem. If f(z) given by () belongs to S b;;gðuþ, then, for any comlex number l,we have (! ) a 2 la 2 6 jbjb max ; B 2 b 2g 2 B The result is shar. 2 g2 g 2 l Proof. The roof is similar to the roof of Theorem. h Proceeding similarly, we now obtain coefficient bounds for the class R b,,g (u). B
Theorem 3. Let u(z) = + B z+b 2 z 2 +B 3 z 3 + If f(z) given by () belongs to R b, (u), then for any comlex number l, a 2 la 2 6 jcj max f; jvjg; ð5þ where v ¼ l bb ð 2Þ B 2 ð Þ 2 B and c ¼ bb 2 ja 3 j 6 jbjb 3 Hðq ; q 2 Þ; ð6þ where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 and q B 2 ¼ B 3 B These results are shar. Proof. A comutation shows that f 0 ðzþ ¼ b z b a z 2 b a 2z 2 3 b a 3z 3 Thus a 2 la 2 ¼ bb 2 w 2 vw 2 ¼ c w2 vw 2 ; ð7þ h i where v ¼ bb lð2þ B ðþ 2 2 B and c ¼ bb. The result now follows from Lemmas 2 and 3. h 2 Remark. When = and uðzþ ¼ Az ð 6 B 6 A 6 Þ; Bz inequality (5) reduces to give the inequality [3, Theorem 4,. 94]. For the class R b,,g (u), we have the following result. Theorem 4. Let u(z)=+b z+b 2 z 2 +B 3 z 3 +. If f(z) given by () belongs to R b, (u), then for any comlex number l, a 2 la 2 6 jcj max f; jvjg; where v ¼ lg2 g 2 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 4 bb ð 2Þ ð Þ 2 B 2 B and c ¼ bb g 2 ð2 Þ ja 3 j 6 jbjb ð3 Þg 3 Hðq ; q 2 Þ; ðþ where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 and q B 2 ¼ B 3 B These results are shar.
42 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 We now rove the coefficient bounds for S ;gða; uþ. Theorem 5. Let u(z) = + B z+b 2 z 2 +B 3 z 3 + Further let r ¼ ð að ÞÞ½ð að ÞÞðB 2 B ÞB 2 Š ; 2B 2 r 2 ¼ ð að ÞÞ½ð að ÞÞðB 2 B ÞB 2 Š ; 2B 2 r 3 ¼ ð að ÞÞ½ð að ÞÞB 2 B 2 Š ; 2B 2 v ¼ 2lB ða 2a Þ B ð að ÞÞ 2 að Þ B 2 If f(z) given by () belongs to S ða; uþ, then >< cv if l 6 r ; a 2 la 2 6 c if r 6 l 6 r 2 ; > cv if l P r 2 B ; c ¼ if r 6 l 6 r 3, then a 2 la 2 ð að ÞÞ2 c ðv Þja 2 B 2 j 2 6 c If r 3 6 l 6 r 2, then a 2 la 2 ð að ÞÞ2 c ð vþja 2 B 2 j 2 6 c For any comlex number l, a 2 la 2 6 c maxf; jvjg B 2ð að 2ÞÞ ð9þ ð20þ B ja 3 j 6 að5 2 Þ3 Hðq ; q 2 Þ; where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 B 2 ðað3 5Þ3Þ B 2ðað ÞÞðað 2ÞÞ and q 2 ¼ B 3 B 2ðað3 5Þ3Þ 2 B 2 ða a Þ B 2ðað ÞÞðað 2ÞÞ These results are shar. ð2þ Proof. If f ðzþ 2S ða; uþ. It is easily shown that að Þ zf 0 ðzþ f ðzþ a z 2 f 00 ðzþ ¼ f ðzþ ð að ÞÞ a z ð3 að5 2 ÞÞ a 3 The roof can now be comleted as in the roof of Theorem. ð að 2ÞÞ 2a 2 ð3 að3 5ÞÞ a a 2 h ð að ÞÞ a 2 z 2 ð að ÞÞ a 3 z 3 ð22þ
For the class S ;gða; uþ, we have the following result. Theorem 6. Let u(z) = + B z+b 2 z 2 +B 3 z 3 +, and let r ¼ g2 ð að ÞÞ½ð að ÞÞðB 2 B ÞB 2 Š 2B 2 g ; 2 r 2 ¼ g2 ð að ÞÞ½ð að ÞÞðB 2 B ÞB 2 Š 2B 2 g ; 2 r 3 ¼ g2 ð að ÞÞ½ð að ÞÞB 2 B 2 Š 2B 2 g ; 2 v ¼ 2g 2lB ða 2a Þ g 2 ð að ÞÞ2 If f(z) given by () belongs to S ;gða; uþ, then >< cv if l 6 r ; a 2 la 2 6 c if r 6 l 6 r 2 ; > cv if l P r 2 if r 6 l 6 r 3, then a 2 la 2 ð að ÞÞ2 g 2 c ðv Þja 2 B 2 j 2 6 c If r 3 6 l 6 r 2, then a 2 la 2 ð að ÞÞ2 g 2 c ð vþja 2 B 2 j 2 6 c For any comlex number l, a 2 la 2 6 c maxf; jvjg R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 43 B að Þ B 2 B ; c ¼ B 2g 2 ð að 2ÞÞ B ja 3 j 6 g 3 ½að5 2 Þ3Š Hðq ; q 2 Þ; where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 B 2 ðað3 5Þ3Þ B 2ðað ÞÞðað 2ÞÞ and q 2 ¼ B 3 B 2ðað3 5Þ3Þ 2 B 2 ða a Þ B 2ðað ÞÞðað 2ÞÞ These results are shar. ð23þ Remark 2. When = and zð 2bÞ uðzþ ¼ ða P 0; 0 6 b < Þ; z (9) and (20) of Theorem 5 reduces to [4, Theorem 4 and 3,. 95]. For the class L M ða; uþ, we now get the following coefficient bounds
44 R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 Theorem 7. Let u(z) = + B z+b 2 z 2 +B 3 z 3 +. Let r ¼ c 3ðB 2 B Þc 2 B 2 c B 2 ; r 2 ¼ c 3ðB 2 B Þc 2 B 2 c B 2 ; r 3 ¼ c 3B 2 c 2 B 2 ; c B 2 c ¼ 4 2 ð 2 2aÞ; c 2 ¼ð aþ½2ð Þ 2 aš2a 3 ; c 3 ¼ 2ð aþ 2 ; v ¼ c l c 2 c 3 B 2 B ; T ¼ 3ð2 ð aþð3 2ÞÞ 3 ; T 2 ¼ a 6 ½ð 6 Þ3 ½6ð aþaðaþš a½ 6 3 2 ð 7aÞ3ð3aÞ3aŠŠ a If f(z) given by () belongs to L M ða; uþ, then 2 B v if l 6 r 2ð2 2aÞ ; >< a 2 la 2 6 2 B if r 2ð2 2aÞ 6 l 6 r 2 ; > 2 B v if l P r 2ð2 2aÞ 2 if r 6 l 6 r 3, then a 2 la 2 c 3 ð vþja j 2 6 B c If r 3 6 l 6 r 2, then a 2 la 2 c 3 ð vþja j 2 6 B c For any comlex number l, a 2 la 2 2 B 6 2ð 2 2aÞ 2 B ja 3 j 6 3ð 3a 3Þ Hðq ; q 2 Þ; where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 B T 4 B 2ð a Þð 2a 2Þ 2 B 2ð 2 2aÞ 2 B 2ð 2 2aÞ ð24þ ð25þ ð26þ max f; jvjg ð27þ and q 2 ¼ B 3 B T 4 ðc 3 B 2 c 2 B 2 Þ 2c 3 ð a Þð 2a 2Þ T 2 6 B 2 ð a Þ 3 These results are shar. Proof. The roof is similar to the roof of Theorem. h Remark 3. As secial cases, we note that for =, inequalities (24) (27) in Theorem 7 are those found in [5, Theorems 2. and 2.2,. 3]. Additionally, if a =, then inequalities (24) (26) are the results established in [4, Theorem 2., Remark 2.2,. 3]. Our final result is on the coefficient bounds for M (a,u).
Theorem. Let u(z) = + B z+b 2 z 2 +B 3 z 3 + Let r ¼ c ðb 2 B Þc 2 B 2 c 3 B 2 ; r 2 ¼ c ðb 2 B Þc 2 B 2 c 3 B 2 c ¼ ð aþ 2 ; c 2 ¼ a ð 2aÞ; c 3 ¼ 2ð 2aÞ; v ¼ ½ð 2aÞð2l Þ ašb B 2 c 3 B If f(z) given by () belongs to M (a,u), then 2 B v if l 6 r 2ð2aÞ ; >< a 2 la 2 6 2 B if r 2ð2aÞ 6 l 6 r 2 ; > 2 B v if l P r 2ð2aÞ 2 ; r 3 ¼ c B 2 c 2 B 2 ; c 3 B 2 if r 6 l 6 r 3, then a 2 la 2 c ð vþja j 2 6 2 B B c 3 2ð 2aÞ ð29þ If r 3 6 l 6 r 2, then a 2 la 2 c ð vþja j 2 6 2 B B c 3 2ð 2aÞ ð30þ For any comlex number l, a 2 la 2 6 2 B max f; jvjg ð3þ 2ð 2aÞ ja 3 j 6 2 B 3ð 3aÞ Hðq ; q 2 Þ; where H(q,q 2 ) is as defined in Lemma 3, q ¼ 2B 2 B R.M. Ali et al. / Alied Mathematics and Comutation 7 (2007) 35 46 45 3ð2 3a 2aÞ 2ð aþð 2aÞ B ; q 2 ¼ B 3 3ð2 3a 2aÞ B 2ð aþð 2aÞ B 2 ð4 5a 3 3 2 að2a Það9a 2Þ2a 2 Þ B 2 2ð aþ 3 ð 2aÞ These results are shar. Proof. For f(z) 2 M (a,u), a comutation shows that a zf 0 ðzþ f ðzþ a zf 00 ðzþ ¼ ð aþa z f 0 ðzþ 3ð 3aÞ 2 ð2 2a aþa 2 3 a 3 3ð2 3a 2aÞ 3 The remaining art of the roof is similar to the roof of Theorem. 2ð 2aÞa 2 2! a a 2 3 3 2 a 3a a a 3 4 Remark 4. When = and uðzþ ¼ð z z Þb ða P 0; 0 < b 6 Þ; (2) and (3) of Theorem reduce to [2, Theorems 2. and 2.2,. 23]. When = and a =,(2) (30) of Theorem reduce to [9, Theorem 3 and Remark,. 7]. h z 2 ð2þ z 3
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