On the Second Hankel Determinant for the kth-root Transform of Analytic Functions

Transcription

1 Filomat 3: DOI 098/FIL707A Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: On the Second Hankel Determinant for the kth-root Transform of Analytic Functions Najla M Alarifi a Rosihan M Ali b V Ravichandran c a Department of Mathematics Imam Abdulrahman Bin Faisal University Dammam 33 Kingdom of Saudi Arabia b School of Mathematical Sciences Universiti Sains Malaysia 800 USM Penang Malaysia c Department of Mathematics University of Delhi Delhi India Abstract Let f be a normalized analytic function in the open unit disk of the complex plane satisfying z f z/ f z is subordinate to a given analytic function ϕ A sharp bound is obtained for the second Hankel determinant of the kth-root transform z [ f z k /z k] k Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes which include the class of α-convex functions and α-logarithmically convex functions These bounds are expressed in terms of the coefficients of the given function ϕ and thus connect with earlier known results for particular choices of ϕ Introduction and Preliminaries Let A denote the class of analytic functions f z = z + a n z n n= defined in the open unit disk D := {z C : z < } For n = and q = the Hankel determinants H q n of the function f A are defined by a n a n+ a n+q a n+ a n+ a n+q H q n := a n+q a n+q a n+q where a = It is evident that H = a 3 a is the Fekete-Szegö coefficient functional of f Interestingly the determinant also satisfies H = S f 0/6 where S f is the Schwarzian derivative of f defined by 00 Mathematics Subject Classification Primary 30C5; Secondary 30C50 30C80 Keywords Second Hankel determinant; subordination; kth-root transformation; Ma-Minda starlike functions; α-convex functions Received: October 0; Accepted: 3 February 05 Communicated by Hari M Srivastava This work benefited greatly from the stimulating discussions held with Dr M H Mohd The work presented here was supported in parts by a research university grant 00/PMATHS/880 from Universiti Sains Malaysia and a grant from University of Delhi addresses: najarifi@hotmailcom Najla M Alarifi rosihan@usmmy Rosihan M Ali vravi@mathsduacin V Ravichandran

2 N M Alarifi et al / Filomat 3: S f = f / f f / f / The Hankel determinants play an importaant role in the study of singularities as well as in the study of power series with integral coefficients [8 5 6] Several earlier investigations include those of [ ] while several recent works are those of [ ] In [5] Lee et al provided a brief survey on the Hankel determinants and obtained bounds for H for functions belonging to several classes defined by subordination In recent years various interesting properties and characteristics including coefficient bounds and coefficient inequalities of many different subclasses of univalent and bi-univalent functions have been investigated The technique used by Ma and Minda for the Fekete-Szego problem for subclasses of convex and starlike functions were used by many authors to solve the same problem for other classes This technique was also used to solve Hankel determinant problem as well as the coefficient problem for bi-univalent functions [ ] The recent works on bi-univalent functions are motivated by the estimates of initial coefficients of certain subclasses of bi-univalent functions by Srivatava et al [50] For a univalent function f of the form the kth-root transform is defined by Fz = [ f z k] k := z [ f z k /z k] k The kth-root transform has been widely used in a variety of ways in complex function theory Since f is univalent whence f z k /z k is non-vanishing in D the kth-root is an analytic function in D Not only does the kth-root transform preserves univalence it is also known [6] to preserve boundedness and starlikeness While the convexity of the kth-root transform of f implies convexity of f the converse however is false [6] In [] Ali et al investigated the Fekete-Szegö coefficient functional for the kth-root transform of functions belonging to several classes defined via subordination It follows from that Fz = z + b kn+ z kn+ n= where the initial coefficients are b k+ = k a b k+ = k a k 3 + a k b 3k+ = k a k k k + a k a 3 + a 3 3!k 3 3 The second Hankel determinant of the associated function Fz = z + n= b kn+z n+ is given by H = b k+ b 3k+ b and this quantity is also known as the second Hankel determinant of the kth-root transform k+ F Recall that an analytic function f is subordinate to an analytic function written f z z if there exists an analytic self-map w of D with w0 = 0 satisfying f z = wz In this paper the best bounds for the second Hankel determinant of the kth-root transform are obtained for several classes of functions defined via subordination These classes can be seen as belonging to the genre of Ma-Minda starlike functions which will be made apparent in the next section The results in this paper are derived through several meticulous lengthy computations and thus in several instances these computations were validated by use of Mathematica Closely related to the classes of functions treated in this paper is the class P consisting of analytic functions with Re pz > 0 in D and normalized by p0 = The following results will be required Lemma [0] If pz = + c z + c z + c 3 z 3 + P then c n Further this bound is sharp

3 N M Alarifi et al / Filomat 3: Lemma [5 p5] If pz = + c z + c z + c 3 z 3 + P then c = c + x c c 3 = c 3 + c c x c c x + c x y 5 for some x y D Another result that will be required is the optimal value of a quadratic expression Standard computations show that LN M max Lt + Mt + N L M > 0 L M 8 N M 0 L M = 0 t 6L + M + N M 0 L M 6 8 or M 0 L M The Second Hankel Determinant of the kth-root Transform This section introduces several classes of normalized analytic functions For each class a sharp bound is obtained for its second Hankel determinant Definition [7] Let ϕ P be given by ϕz = + B z + B z + B 3 z 3 + B > 0 z D 7 Further assume that ϕ maps D onto a region starlike with respect to and ϕd is symmetric with respect to the real axis The class S ϕ consists of functions f A satisfying z f z/ f z ϕz In the literature this class S ϕ is widely called the Ma-Minda starlike functions with respect to ϕ For the particular case when ϕ is given by ϕ α z := + αz z = + αz + αz + αz α < the class S α := S ϕ α is the well-known class of starlike functions of order α For the function ϕ PAR z := + log + z π = + 8 z π z + 6 3π z + 8 5π z3 + S ϕ PAR is the class S p of parabolic starlike functions introduced by Rønning [8]: { z f S z z f } z p := f A : Re > f z f z Ali and Ravichandran [] gave a survey on parabolic starlike functions and its related class of uniformly convex functions When + z β ϕ β z := = + βz + β z + z 3 β + β z < β the class S ϕ β is the familiar class S of strongly starlike functions of order β: β { S β := f A : z f arg z f z < βπ } The class S + z is the class of lemniscate of Bernoulli starlike functions studied in [9]: S L := f A : z f z f z <

4 N M Alarifi et al / Filomat 3: Theorem Let ϕ be given by 7 f z = z + n= a n z n S ϕ and Fz = z + n= b kn+z kn+ be its kth-root transform Further let δ = /k If B B and B 3 satisfy the conditions B B and B B 3 δb 3B 3B 0 If B B and B 3 satisfy the conditions or the conditions 3 If B B and B 3 satisfy the conditions H = b k+ b 3k+ b k+ B k B B and B B 3 δb 3B B B B 0 B B and B B 3 δb 3B 3B 0 b k+ b 3k+ b k+ k B B 3 δb 3B B B and B + B B B B 3 δb 3B 0 b k+ b 3k+ b k+ B 3 B B 3 δb k 3B B B B B B B 3 δb 3B B B B Proof Since f S ϕ there exists an analytic self-map w of D with w0 = 0 satisfying z f z f z = ϕwz 8 Define the function P P by or equivalently wz = P z P z + = c z + P z = + wz wz = + c z + c z + c c z + By using 9 along with 7 lead to the expansion c 3 c c + c3 ϕwz = + B wz + B w z + = + B c z + B c 3 + B c c + c3 8 + B c c c + B 3c 3 8 z3 + c c 9 + B c z z Now z f z f z = + a z + a 3 a z + 3a 3a a 3 + a 3 z 3 +

5 and comparing with 8 and 0 it follows that N M Alarifi et al / Filomat 3: a = B c a 3 = B 8 B + B c + B c a = B + B + B 3 8 3B + 3B B + B 3 c 3 + 3B B + B c c + 8B c 3 Consequently and 3 yield b k+ = B c k b k+ = B 8k B + B c + B k c + B 8k c b 3k+ = B 3 8k 3B + B B + 3B B + B 3 c 3 + 3B B + B c c kb c [ + 8B c 3 + B 6k B + B c + B ] k k c + B 3 8k 3 c3 A lengthy computations validated by Mathematica show that b k+ b 3k+ b k+ = B B 9k B c c + B k B3 B + B 3 3 B c B + 6B c c 3 B c Next for ease in computations let d = 6B d = B B d 3 = B d = B δb 3 B + B 3 3 B B and T = B 9k Then b k+ b 3k+ b k+ = T d c c 3 + d c c + d 3 c + d c 3 Since the function p e iθ z θ R is in the class P for any p P there is no loss of generality in assuming c = c > 0 c [0 ] Substituting the values of c and c 3 respectively from and 5 in 3 it follows that b k+ b 3k+ b k+ = T c d + d + d 3 + d + xc c d + d + d 3 + c x d c + d 3 c + d c x y for some x y D With s = x yields b k+ b 3k+ b k+ T c 6B 3 δb 3 B B + 8sc c B + s c B c + 8B + 3B c c s = T c B 3 δb 3 3B B + 8B c c + B sc c + B s c c c 6 := Fc s

6 N M Alarifi et al / Filomat 3: c s [0 ] [0 ] Now F s = T B c c + B s c c c 6 and so F/ s > 0; that is Fc s is an increasing function of s Hence max Fc s = Fc := Gc 0 s Upon simplification we find that Gc = B c B 9k 3 δb 3 3B B B + 8c B B + 8B Writing c = t and L = B 3 δb 3 3B B B M = 8 B B N = 8B B it follows from 6 that LN M 9k bk+ b 3k+ b L M > 0 L M 8 B k+ N M 0 L M 6L + M + N M 0 L M 8 or M 0 L M These conditions now lead to the desired bounds for the second Hankel determinant Remark 3 The special case k = in Theorem reduces to [5 Theorem ] If k = and B = B = B 3 = then Theorem reduces to [3 Theorem 3] Judicious choices of ϕ in Theorem lead to the following results for the special cases Corollary If f S α then b k+ b 3k+ b k+ α /k If f S L then b k+b 3k+ b k+ /6k 3 If f S P then b k+b 3k+ b k+ 6/π k If f S β then b k+b 3k+ b k+ β /k B Definition 5 Let ϕ P be given by Definition and b be a non-zero complex number The class R b ϕ consists of functions f A satisfying the subordination + f z ϕz b This class was considered in [5] for the more general case of p-valent functions The case b = and ϕz = + z/ z gives the subclass of close-to-convex functions studied by MacGregor [8] consisting of functions whose derivative has positive real part Al Amiri et al [] introduced the general class of analytic functions satisfying Re { + /b z f z/ z } > 0 for some starlike function It is evident that R b ϕ coincides with this class for z = z and ϕz = + z/ z Theorem 6 Let ϕ be given by 7 f z = z + n= a n z n R b ϕ and Fz = z + n= b kn+z kn+ be its kth-root transform Further let λ = 9B B 3 8B + δb and δ = 3 k b 3 k

7 If B B and B 3 satisfy the conditions If B B and B 3 satisfy the conditions or the conditions N M Alarifi et al / Filomat 3: B 7 B and 8B λ 0 b k+ b 3k+ b k+ B b 9k B 7 B and 3 If B B and B 3 satisfy the conditions 9 B + B B λ 0 B 7 B and 8B λ 0 b k+ b 3k+ b k+ b λ 3 3 k B 7 B and 9 B + B B λ 0 b k+ b 3k+ b k+ b B 3λ 36B B 8B 5 3 k B λ B B B Proof The proof is similar to Theorem There exists an analytic self-map w of D satisfying + f z = ϕ wz b Now + f z = + b b a z + 3 b a 3z + b a z 3 + and comparing with 0 and we find that a = b B c B c B c a 3 = b + B c B c 3 B c c + B c 3 + B c c B c 3 + B 3c 3 a = b Consequently 5 and 3 yield 3 5 b k+ = b B c k b k+ = b B c B c k + B c kb + B 3k c b 3k+ = b B c 3 B c c + B c 3 3k + B c c B c 3 + B 3c 3 kb + B 8k c c B c3 + B B c 3 k k + b 3 B 3 38 c3

8 N M Alarifi et al / Filomat 3: Thus B b k+ b 3k+ b k+ = b 7 3 k B B + 3 B B 3 3 B + 3 k b B 3 k c + B B B c c + 3 B c c 3 5 B c Writing d = 36B d = B B B d 3 = 3B d = B B B + 9B B 3 8B + δb b and T = 7 3 k 6 then bk+ b 3k+ b Equations and 5 show that k+ = T d c c 3 + d c c + d 3 c + d c b k+ b 3k+ b k+ = T c d + d + d 3 + d + xc c d + d + d 3 + c x d c + d 3 c + d c c x y for some x y D With s = x 6 yields b k+ b 3k+ b k+ T c λ + sc c B B + c s B 9c + 8 c 8c + 8B c c s c s [0 ] [0 ] Now = T c λ + sc c B B + 8B c c + c s B c c 6 := Fc s F s = T c c B B + s c B c c 6 and so F/ s > 0; that is max Fc s = Fc := Gc 0 s Routine simplifications yield Gc = T c λ B B B + c B B 7B + 8B With c = t and L = λ B B B M = B B 7B N = 8B it follows from 6 that LN M 7 3 k bk+ b b 3k+ b L M > 0 L M 8 k+ N M 0 L M 6L + M + N M 0 L M 8 or M 0 L M Inserting the values of the parameters L M and N yield the desired conditions and bounds

9 Theorem 6 yields the following special cases Remark 7 N M Alarifi et al / Filomat 3: With k = and b = τ Theorem 6 reduces to [5 Theorem 3] for γ = 0 With k = and b = Theorem 6 reduces to [ Theorem ] for γ = 0 3 For the choice ϕz = + Az/ + Bz B < A k = and b = τ Theorem 6 reduces to [7 Theorem ] for γ = 0 We next introduce a third class of functions to be studied Definition 8 Let ϕ P be given by Definition and α 0 The class S α ϕ consists of functions f A satisfying the subordination z f z f z + α z f z f z ϕz Padmanabhan [] introduced the class S α ϕ in 00 and investigated sufficient conditions for starlikeness The special case when α = and ϕ = + z/ z was considered in [7] It is evident that S 0 ϕ reduces to the class S ϕ treated in Definition Theorem 9 Let ϕ be given by 7 f z = z + n= a n z n S α ϕ and Fz = z + n= b kn+z kn+ be its kth-root transform Further let λ = δb αb B + + α + 3α 3 B B α + αb u = + αα + 6α + and δ = 0α + 5k 33 α + k 3 α 3 + αk If B B and B 3 satisfy the conditions u B + α + 6α + 6α B + αb 0 and λ 3 + α + αb 0 If B B and B 3 satisfy the conditions b k+ b 3k+ b B k+ + 3αk u B + α + 6α + 6α B + αb 0 and λ B u B + αb + α + 6α + 9α B 0 or the conditions u B + α + 6α + 6α B + αb 0 and λ 3 + α + αb 0 b k+ b 3k+ b λ k+ 3 + α + α + 3αk

10 3 If B B and B 3 satisfy the conditions N M Alarifi et al / Filomat 3: u B + α + 6α + 6α B + αb 0 and λ B u B + αb + α + 6α + 9α B 0 b k+ b 3k+ b k+ B 3 + α + 6α + 8α λ + u + αb B + ub B αb + α + 6α + 6α B + u B / k + α + 3α + α λ B u + αb ub B Proof Let z f z + α z f z f z for an analytic self-map w of D Since z f z + α z f z f z it follows from 0 7 and 8 that = ϕ wz 7 = + + αa z + + 3αa 3 + αa αa 3 + 8αa a αa 3 z a = + α B c a 3 = + α B c 3 B B c + α + 3α + B c a = 8 B c 3 B c c + B c c B c 3 + α 3 + B c 3 + B 3c α + α + 3α Consequently 3 yields + α B c c B c3 + B B 3 B c 3 + B 3 c3 c3 + α B c b k+ = k + α b k+ = B c 3 + B B c 3 kα + k + 3α + 8k + α + 3α B c b 3k+ = B c 3 3 B c c + B c c B c 3k + α 3 + B c 3 + B 3c 3 α3 k B 3k + α + 3α + α c c B c3 + B B c 3 α + 7α + αk α3 k + 3 z + 3k 3 + α 3 + 3α + α B 3 c3

11 Routine computations show that N M Alarifi et al / Filomat 3: α 3 b k+ b 3k+ b k+ = k + 5α k 36α 3 33α 0α B 6 3k + α + 3α + α c αk + B 5 3k 3 + α + 3α + α B B c B3 c c α + 6α + which yields where + B B 6 3k + α + 3α B c + α + B B B c c + B 3k + α + α c c 3 + B B 3 c B 6 k + 3α c + B c b k+ b 3k+ b k+ = T B ub B + + α + 3α B 3 T = By writing + αb B B + δb α + αb + B ub B αb c c + 6B + α + 3α c c 3 B + α + αc 6 3k + α + 3α + α d = 6B + α + 3α d = B ub B αb d 3 = B + α + α d = B ub B + + α + 3α B 3 + αb B B + δb α + αb 9 c then b k+ b 3k+ b k+ = T d c c 3 + d c c + d 3 c + d c Consequently b k+ b 3k+ b k+ = T c d + d + d 3 + d + xc c d + d + d 3 + d + c x d c + d 3 c + d c c x y

12 for some x y D With s = x 9 yields where N M Alarifi et al / Filomat 3: b k+ b 3k+ b k+ T c λ + B sc c u B + αb + s + α c + 3α B c α + αb c α + 3α B c c s = T c λ + sb c c u B + αb α + 3α B c c + s + α + 6α + α c B c c p := Fc s p = 6 + 6α + 8α + 6α + α > Proceeding similarly as in the previous proofs it can be shown that Fc s is an increasing function of s whence max Fc s = Fc := Gc 0 s with Gc = T c λ B u B + B + αb + 8B c u B + α + 6α + 6α B + αb α + 6α + 8α B Writing c = t and 6 yield L = λ B u B + B + αb M = 8B u B + α + 6α + 6α B + αb N = 8 + α + 6α + 8α B 0 LN M b k+ b 3k+ b L M > 0 L M 8 k+ T N M 0 L M 6L + M + N M 0 L M 8 or M 0 L M where L M N are given by 0 This completes the proof Definition 0 Let ϕ P be given by Definition and α [0 ] The class Lα ϕ consists of functions f A satisfying the subordination z f α z + z f α z f z f ϕz z This class is analogous to the α-logarithmically convex functions introduced by Lewandowski et al [6] In [] Darus et al found sharp upper bounds for a a 3 and a 3 µa µ real for f Lα ϕ Evidently L ϕ reduces to the class S ϕ treated in Definition

13 N M Alarifi et al / Filomat 3: Theorem Let ϕ be given by 7 f z = z + n= a n z n Lα ϕ and Fz = z + n= b kn+z kn+ be its kth-root transform Further let λ = δb + vb B + α3 α B B 3 3 α 3αB u = α 7α 8α + v = 7α 3 3α 6α + and δ = α 3 k α 5 7α + 6α α 3 9k 8 0α α + 3α 3 6 7α + 6α + 93α α 3 + k + α + α / 3 k α If B B and B 3 satisfy the conditions vb + u B α 6 6α + α B 0 and λ + 3 α 3α B 0 If B B and B 3 satisfy the conditions b k+ b 3k+ b B k+ 3 αk vb + u B α 6 6α + α B 0 and λ B vb + u B + α3 α B 0 or the conditions vb + u B α 6 6α + α B 0 and λ + 3 α 3α B 0 3 If B B and B 3 satisfy the conditions b k+ b 3k+ b λ k+ 3 3α α3 αk vb + u B α 6 6α + α B > 0 and λ B vb + u B + α3 α B 0 b k+ b 3k+ b k+ B vb + α 6 6α + α B + u B + α 3α ub + vb3 λ + ub B / 3 3α α3 αk λ ub vb3 ub B Proof With z f α z + z f α z f z f = ϕ wz z

14 the above equation and 0 yield N M Alarifi et al / Filomat 3: a = B c α a 3 = 3 α α + 5α 8 B α c + B c B c + B c a = B c 3 c B c 3 3 3α + B 3c 3 + B 3c 3 + B c 3 B c c + B c 3 α 3 + α + 0α 8 α + α α 3 3α B3 c α 3 3 α 3α α + 5α 8 B 3 c3 α B c c + α B c3 α B B c 3 From and 3 and after some lengthy computations validated by Mathematica we find that b k+ b 3k+ b k+ = T δb + ub B B + vb B B + α3 α B B 3 3 α 3αB c + B u B B + vb c c + 6B α3 α c c 3 B α 3αc where Let T = 3 3α 3 α3 αk d = 6B α3 α d = B u B B + vb d 3 = B α 3α d = δb + ub B B + vb B B + α3 α B B 3 3 α 3αB Then bk+ b 3k+ b k+ = T d c c 3 + d c c + d 3 c + d c Thus b k+ b 3k+ b k+ = T c d + d + d 3 + d + xc c d + d + d 3 + c x d c + d 3 c + d c c x z T c λ + sc c B u B + vb + s c αb 7α 8α + c + α 3α + 8c c α3 α B s = T c λ + sc c B u B + vb + 8c c α 3 α B + s c α 8α + 7α B c c p := Fc s

15 where N M Alarifi et al / Filomat 3: p = 6 8 0α + 3α 8α + 7α > The function Fc s is increasing relative to s and so max Fc s = Fc := Gc 0 s with Gc = T c λ B vb + u B + B + c B u B + vb α α 6α + 6 B + 8 α 3αB Letting c = t and L = λ B vb + u B + B M = B u B α α 6α + 6 B + vb N = 8 α 3αB 6 leads to LN M b k+ b 3k+ b L M > 0 L M 8 k+ T N M 0 L M 6L + M + N M 0 L M 8 or M 0 L M where L M N are given by The proof is now evident Remark With α = Theorem reduces to Theorem If k = and α = 0 then Theorem reduces to [5 Theorem ] 3 With the choice of ϕz = + z/ z k = and α = 0 Theorem reduces to [3 Theorem 3] We conclude with the following class of functions Definition 3 Let ϕ P be given by Definition and α 0 The class Mα ϕ consists of functions f A satisfying the subordination α z f z + α + z f z f z f ϕz z The class Mα ϕ is analogous to the α-convex functions of Mocanu et al[30] who investigated geometric properties of the class in the case ϕ = + z/ z Theorem Let ϕ be given by 7 f z = z + n= a n z n Mα ϕ and Fz = z + n= b kn+z kn+ be its kth-root transform Further let λ = δb + 6αB B + + α B B α + 3αB u = 7α + α + and δ = k α k α + + k α 3 + α 3 k

16 If B B and B 3 satisfy the conditions N M Alarifi et al / Filomat 3: u B + 3αB α + α + B 0 and λ 3 + α + 3αB 0 If B B and B 3 satisfy the conditions b k+ b 3k+ b B k+ + αk u B + 3αB α + α + B 0 and λ B + α B + 3αB + u B 0 or the conditions u B + 3αB α + α + B 0 and λ 3 + α + 3αB 0 3 If B B and B 3 satisfy the conditions b k+ b 3k+ b λ k+ 3 + α + 3α + αk u B + 3αB α + α + B > 0 and λ B + α B + 3αB + u B 0 b k+ b 3k+ b k+ B 6αB + α + α B + u B 3 + α + 3α λ + u + αb B + ub B / 3 + α + 3α + αk λ u + αb B ub B Proof For f Mα ϕ direct lengthy computations reveal that b k+ b 3k+ b k+ = T δb + 6αB B B + ub B B + + α B B α + 3αB c + B 3αB + u B B c c + 6B + α c c 3 B + α + 3αc where T = By writing 3 + α + 3α 3 + αk d = 6 + α B d = B 3αB + u B B d 3 = + α + 3αB d = δb + 6αB B B + ub B B + + α B B α + 3αB

17 then bk+ b 3k+ b Consequently where N M Alarifi et al / Filomat 3: k+ = T d c c 3 + d c c + d 3 c + d c b k+ b 3k+ b k+ = T c d + d + d 3 + d + xc c d + d + d 3 + c x d c + d 3 c + d c c x y T c λ + sc c B 3αB + u B + s c B uc + + α + 3α + 8c c + α B s = T c λ + sc c B 3αB + u B + 8c c + α B + s c ub c c p := Fc s p = 6 + α + 3α u The function Fc s is an increasing function of s Thus > max Fc s = Fc := Gc 0 s with Gc = T c λ B 6αB + u B + B + c B 6αB + u B + α + α B α + 3αB Next let c = t and L = λ B 6αB + u B + B M = 8B 3αB + u B + α + α B N = 8 + α + 3αB 3 From 6 it follows that LN M b k+ b 3k+ b L M > 0 L M 8 ; k+ T N M 0 L M ; 6L + M + N M 0 L M 8 or M 0 L M where L M N are given by 3 The rest of the proof is now evident Theorem yields the following special cases Remark 5 If α = 0 then Theorem reduces to Theorem When k = and α = Theorem reduces to [5 Theorem ] 3 Choosing ϕz = + z/ z k = and α = then Theorem reduces to [3 Theorem 3]

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