Coefficient determinants involving many Fekete-Szeg O ¨ -type parameters
aa r X i v : . [ m a t h . C V ] N ov COEFFICIENT DETERMINANTS INVOLVING MANYFEKETE-SZEG ¨ O -TYPE PARAMETERS K. O. BABALOLA
Abstract.
We extend our definition (in a recent paper [3]) of the coeffi-cient determinants of analytic mappings of the unit disk to include manyFekete-Szeg¨ o -type parameters, and compute the best possible bounds oncertain specific determinants for the choice class of starlike functions. AMS Mathematics Subject Classification (2010) : 30C45, 30C50.
Keyword : Hankel determinants, coefficient determinants, Fekete-Szeg¨ o parame-ters, starlike functions. Hankel Determinants with Parameters
In a recent paper [3], we extended the definition of the well known Hankeldeterminant for coefficients of analytic mappings to include the also wellknown Fekete-Szeg¨ o parameter as follows: Definition A.
Let λ be a nonnegative real number. Then for integers n ≥ q ≥
1, we define the q -th Hankel determinants with Fekete-Szeg¨ o parameter λ , that is H λq ( n ), as H λq ( n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n a n +1 · · · λa n + q − a n +1 · · · · · · ...... ... ... ... a n + q − · · · · · · a n +2( q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) with the well known Hankel determinant being the case λ = 1.We also defined other similar determinants as: Definition B.
Let λ be a nonnegative real number. Then for integers n ≥ q ≥
1, we define the B λq ( n ) determinants as B λq ( n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n a n +1 · · · a n + q − a n + q a n + q +1 · · · a n +2 q − a n +2 q a n +2 q +1 · · · a n +3 q − ... ... ... ... a n + q ( q − · · · · · · λa n + q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Our motivation for those new definitions lie in the fact that, for functionclasses defined by other function classes (for example the classes of close-to-star, close-to-convex, quasi-convex, α -starlike, α -convex, α -close-to-star, α -close-to-convex whose definitions involve other function classes), coefficientfunctionals of the form | a a − λa | and | a a − λa | (and possibly more) forthe defining function classes have frequently appeared to be resolved in theinvestigations of Hankel determinants for the desired classes of functions.In this paper, we further extend these definitions to include finitely manyFekete-Szeg¨ o parameters λ j , j = 1 , , · · · in order to accomodate a widevariety of emerging functionals in the study of coefficients of mappings ofthe unit disk. Now we say: Definition 1.
Let λ i , i = 1 , , · · · , q be nonnegative real numbers. Thenfor integers n ≥ q ≥
1, we define the q -th Hankel determinants withFekete-Szeg¨ o parameters λ i , that is H λ ,λ , ··· ,λ q q ( n ), as H λ ,λ , ··· ,λ q q ( n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ a n λ a n +1 · · · λ q a n + q − a n +1 · · · · · · ...... ... ... ... a n + q − · · · · · · a n +2( q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and we also say: Definition 2.
Let λ j , j = 1 , , · · · , n be nonnegative real numbers. Thenfor integers n ≥ q ≥
1, we define the B λ ,λ , ··· ,λ n q ( n ) determinants as B λ ,λ , ··· ,λ n q ( n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n a n +1 · · · λ a n + q − a n + q a n + q +1 · · · λ a n +2 q − a n +2 q a n +2 q +1 · · · λ a n +3 q − ... ... ... ... a n + q ( q − · · · · · · λ n a n + q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For λ j = 1, j = 1 , , · · · , q − n − H λq ( n ) and B λq ( n ) inplace of H , , ··· , ,λ q q ( n ) and B , , ··· , ,λ n q ( n ) respectively, and we quickly notethat for real numbers, γ , α and β we have: H γ (1) = (cid:12)(cid:12)(cid:12)(cid:12) γa a a (cid:12)(cid:12)(cid:12)(cid:12) = a − γa ,H α (2) = (cid:12)(cid:12)(cid:12)(cid:12) a αa a a (cid:12)(cid:12)(cid:12)(cid:12) = a a − αa and B β (1) = (cid:12)(cid:12)(cid:12)(cid:12) a a βa (cid:12)(cid:12)(cid:12)(cid:12) = a a − βa . Then, in this paper we shall investigate the determinant H λ ,λ ,λ (1)for the favoured and well known class of starlike functions, for which Re OEFFICIENT DETERMINANTS INVOLVING MANY FEKETE-SZEG ¨ O -TYPE PARAMETERS3 zf ′ ( z ) /f ( z ) >
0, denoted by S ∗ . By definition H λ ,λ ,λ (1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ a λ a λ a a a a a a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For f ∈ S ∗ , a = 1 so that H λ ,λ ,λ (1) = a ( λ a a − λ a ) − a ( λ a − λ a a ) + a ( λ a − λ a )= λ a ( a a − αa ) + λ a ( a a − βa ) + λ a ( a − γa )where α = λ /λ , β = λ /λ and γ = λ /λ . Thus by triangle inequality,we have | H λ ,λ ,λ (1) | ≤ λ | a || H γ (1) | + λ | a || H α (2) | + λ | a || B β (1) | . (1 . o parametersin this paper, which leads to ease of analysis and better precision. For ourchoice class, S ∗ , the Fekete-Szeg¨ o functional, | H γ (1) | , is known and givenas: Theorem 1 ([6]) . Let f ∈ S ∗ . Then for real number γ , | H γ (1) | ≤ max { , | γ − |} . The inequality is sharp. For each γ , equality is attained by f ( z ) given by f ( z ) = ( z − z i f ≤ γ ≤ , z (1 − z ) i f γ ∈ (cid:2) , (cid:3) ∪ [1 , ∞ ) . Thus we shall in this paper obtain the best possible bounds on H α (2)and B β (1) to conclude our investigation. In the next section we state thelemmas we shall use to establish the desired bounds in Section 3.2. Preliminary Lemmas
Let P denote the class of functions p ( z ) = 1 + c z + c z + · · · which areregular in E and satisfy Re p ( z ) > z ∈ E . To prove the main results inthe next section we shall require the following two lemmas. Lemma 1 ([4]) . Let p ∈ P , then | c k | ≤ , k = 1 , , · · · , and the inequality issharp. Equality is realized by the M ¨ o bius function L ( z ) = (1 + z ) / (1 − z ) . Lemma 2 ([1]) . Let p ∈ P . Then (cid:12)(cid:12)(cid:12)(cid:12) c − σ c (cid:12)(cid:12)(cid:12)(cid:12) ≤ − σ ) , σ ≤ , , ≤ σ ≤ , σ − , σ ≥ { , | σ − |} . The inequality is sharp. For each σ , equality is attained by p ( z ) given by p ( z ) = ( z − z i f ≤ σ ≤ , z − z i f γ ∈ [ −∞ , ∪ [2 , ∞ ) . K. O. BABALOLA
Lemma 3 ([7]) . Let p ∈ P , then c = c + x (4 − c ) (2 . and c = c + 2 xc (4 − c ) − x c (4 − c ) + 2 z (1 − | x | )(4 − c ) (2 . for some x , z such that | x | ≤ and | z | ≤ . Inequalities for | H λ ,λ ,λ (1) | of S ∗ First we prove:
Theorem 2.
Let f ∈ S ∗ . Then for real number β , | B β (1) | ≤ { β, | − β |} . The inequalities are sharp. For each β , equality is attained by f ( z ) given by f ( z ) = ( z (1 − z ) i f ≤ β ≤ , z (1 − z ) i f β ∈ [0 , ∪ [3 , ∞ ) . Proof.
It is well known that if f ∈ S ∗ , then a = c , 2 a = c + c and6 a = 2 c + 3 c c + c . Thus we have | B β (1) | = | a a − βa | = (cid:12)(cid:12)(cid:12)(cid:12) − β c + 1 − β c c − β c (cid:12)(cid:12)(cid:12)(cid:12) . (3 . β ≤
1, then (3.1) yields | B β (1) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) − β c − β c (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − β c c (cid:12)(cid:12)(cid:12)(cid:12) which, by Lemma 1 gives | B β (1) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) − β c − β c (cid:12)(cid:12)(cid:12)(cid:12) + 2(1 − β ) . (3 . c using Lemma 3, we have (cid:12)(cid:12)(cid:12)(cid:12) − β c − β c (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (2 − β ) c − βc (4 − c ) x βc (4 − c ) x − β (4 − c )(1 − | x | ) z (cid:12)(cid:12)(cid:12)(cid:12) . By Lemma 1 again, | c | ≤
2. Then letting c = c , we may assume withoutrestriction that c ∈ [0 ,
2] so that (cid:12)(cid:12)(cid:12)(cid:12) − β c − β c (cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 − β ) c β (4 − c )6 + βc (4 − c ) ρ β ( c − − c ) ρ F ( ρ ) . OEFFICIENT DETERMINANTS INVOLVING MANY FEKETE-SZEG ¨ O -TYPE PARAMETERS5 Since β ≤
1, the extreme points of F ( ρ ) are ρ = 0, ρ = 1 and ρ = c/ (2 − c )(with c ∈ [0 ,
1] in this case since ρ ∈ [0 , G ( c ) = F (0) = (2 − β ) c β (4 − c )6 G ( c ) = F (1) = (2 − β ) c β (4 − c )6 + βc (4 − c )6 + β ( c − − c )12= (1 − β ) c βc and G ( c ) = F (cid:18) c − c (cid:19) = (2 − β ) c β (4 − c )6 + βc ( c + 2)12= (3 − β ) c β , c ∈ [0 , . By elementary calculus, we find that G ( c ) ≤ G ( c ) ≤ G c (2) = 4 − β while G ( c ) ≤ G (1) = (1 + β ) /
2. Hence for β ≤ − β . Using this in (3.3) we have | B β (1) | ≤ − β for 0 ≤ β ≤ ≤ β ≤
3. Then we write (3.1) | B β (1) | = (cid:12)(cid:12)(cid:12)(cid:12) β c β − c c − β − c (cid:12)(cid:12)(cid:12)(cid:12) which gives | B β (1) | ≤ β | c | β − | c | (cid:12)(cid:12)(cid:12)(cid:12) c − β − β − c (cid:12)(cid:12)(cid:12)(cid:12) . Observing that ( β − / (3 β − ≤ | B β (1) | ≤ β for 1 ≤ β ≤ β ≥ | B β (1) | = (cid:12)(cid:12)(cid:12)(cid:12) β c β − c c + β − c (cid:12)(cid:12)(cid:12)(cid:12) and apply Lemma 1 to obtain | B β (1) | ≤ β − | B β (1) | ≤ − β i f ≤ β ≤ , β i f ≤ β ≤ , β − f β ≥ . This completes the proof. (cid:3)
Corollary 1 ([2]) . Let f ∈ S ∗ . Then | B (1) | ≤ . The inequality is sharp. Equality is attained by the Koebe function k ( z ) = z/ (1 − z ) . K. O. BABALOLA
Next we have
Theorem 3.
Let f ∈ S ∗ . Then for real number α , | H α (2) | ≤ max { , | α − |} . The inequalities are sharp. For each α , equality is attained by f ( z ) given by f ( z ) = ( z − z / √ α i f < α ≤ , z (1 − z ) i f α ∈ (cid:2) , (cid:3) ∪ [1 , ∞ ) . Proof.
Following from the proof of Theorem 2, if f ∈ S ∗ , then a = c ,2 a = c + c and 6 a = 2 c + 3 c c + c . Thus we have | H α (2) | = | a a − αa | = (cid:12)(cid:12)(cid:12)(cid:12) c c − α ) c c − α ) c − α c (cid:12)(cid:12)(cid:12)(cid:12) . (3 . α ≤ /
3. Then we have | H α (2) | ≤ | c || c | − α ) | c |
12 + α | c | (cid:12)(cid:12)(cid:12)(cid:12) c − − α ) α c (cid:12)(cid:12)(cid:12)(cid:12) which, by Lemmas 1 and 2 noting that 4(1 − α ) /α ≥ α ≤ /
3, gives | H α (2) | ≤ − α .Substituting for c and c in (3.3) using Lemma 3 we obtain | H α (2) | = (cid:12)(cid:12)(cid:12)(cid:12) (8 − α ) c
16 + (10 − α ) c (4 − c ) x − c (4 − c ) x c (1 − | x | )(4 − c ) z − α (4 − c ) x (cid:12)(cid:12)(cid:12)(cid:12) (3 . | H α (2) | = (cid:12)(cid:12)(cid:12)(cid:12) (9 α − c − (10 − α ) c (4 − c ) x
24 + c (4 − c ) x − c (1 − | x | )(4 − c ) z α (4 − c ) x (cid:12)(cid:12)(cid:12)(cid:12) . Letting c = c , assuming without restriction that c ∈ [0 ,
2] and setting ρ = | x | , we have | H α (2) | ≤ (9 α − c
16 + c (4 − c )6 + (10 − α ) c (4 − c ) ρ − c )( c − − α ) c − α ] ρ
48 = F ( ρ ) . Then F ′ ( ρ ) = (10 − α ) c (4 − c )24 + (4 − c )( c − − α ) c − α ] ρ / ≤ α ≤ /
9. Hence for 2 / ≤ α ≤ / F ( ρ ) is increasing on [0 ,
1] so that F ( ρ ) ≤ F (1). Thus we have | H α (2) | ≤ F (1) = ( α − c − α − c + α = G ( c ) OEFFICIENT DETERMINANTS INVOLVING MANY FEKETE-SZEG ¨ O -TYPE PARAMETERS7 Now by elementary calculus we see that the maximum of G ( c ) on [0 ,
2] occursat c = 1 if 2 / ≤ α ≥ c = 2 for 1 ≤ α ≤ / G (1) = 1 for 2 / ≤ α ≤ G (2) = 9 α − ≤ α ≤ / α ≥ /
9. Then we write (3.4) as | H α (2) | = (cid:12)(cid:12)(cid:12)(cid:12) (9 α − c
16 + (9 α − c (4 − c ) x
24 + c (4 − c ) x − c (1 − | x | )(4 − c ) z α (4 − c ) x (cid:12)(cid:12)(cid:12)(cid:12) . By similar argument as above we have | H α (2) | ≤ F (1) = 3 α − c + 3 α − c + α = G ( c )which yields | H α (2) | ≤ α − α ≥ | H α (2) | ≤ α − | H α (2) | ≤ − α i f ≤ α ≤ , f < α ≤ , α − f α ≥ (cid:3) Corollary 2 ([5]) . Let f ∈ S ∗ . Then | H (2) | ≤ . The inequality is sharp. Equality is attained by the Koebe function k ( z ) = z/ (1 − z ) . Now using Theorems 1 to 3 in (1.2) and setting α = λ /λ , β = λ /λ and γ = λ /λ , we obtain the grand inequalities for | H λ ,λ ,λ (1) | as follows: Theorem 4.
Let f ∈ S ∗ . Then for real numbers λ j , j = 1 , , , | H λ ,λ ,λ (1) | ≤ { λ , | λ − λ |} + 8 max { λ , | λ − λ |} + 3 max { λ , | λ − λ |} . The inequalities are sharp.
We conclude with the following interesting and nice bounds on coefficientdeterminants of starlike functions.
Corollary 3.
Let f ∈ S ∗ . Then | H , , (1) | ≤ , | H , , (1) | ≤ , | H , , (1) | ≤ . , | H , , (1) | ≤ , | H , , (1) | ≤ , | H , , (1) | ≤ , | H , , (1) | ≤ . , | H , , (1) | ≤ . All inequalities are best possible in the sense that component inequalities inthe last theorem are best possible.
The first | H (1) | = | H , , (1) | ≤
16 was reported in [2].
K. O. BABALOLA
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