The Fekete–Szegö problem for k-uniformly convex functions and for a

J. Math. Anal. Appl. 347 (2008) 563–572

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The Fekete–Szegö problem for k-uniformly convex functions and for a class defined by the Owa–Srivastava operator A.K. Mishra ∗,1 , P. Gochhayat Berhampur University, Department of Mathematics, Bhanja Bihar, 760007 Berhampur, Ganjam, Orissa, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 January 2007 Available online 12 June 2008 Submitted by H.M. Srivastava

By making use of a fractional differential operator due to Owa and Srivastava, a subclass of analytic functions related to k-uniformly convex functions, is introduced. For this class and in particular for the class of k-uniformly convex functions the Fekete–Szegö problem is completely solved. © 2008 Elsevier Inc. All rights reserved.

Keywords: k-Uniformly convex function Elliptic integral Hadamard product Fractional derivative Fekete–Szegö problem

1. Introduction and definitions Let A denote the family of functions analytic in the open unit disk

  U := z: z ∈ C and | z| < 1

and let A0 be the class of functions f in A given by the normalized power series f ( z) = z +

∞ 

an zn

(z ∈ U ).

(1.1)

n=2

Let S denote the class of functions which are univalent in U . For fixed k (0  k < ∞), the function f ∈ A0 is said to be in k-U CV ; the class of k-uniformly convex functions in U , if the image of every circular arc γ contained in U , with centre ξ where |ξ |  k, is a convex arc. This interesting unification of the concepts of convex functions (cf. [4]) and uniformly convex functions (cf. [6]) is due to Kanas and Wisniowska [8]. The class of k-SP is defined from k-U CV via the Alexander’s transform (see [9]), i.e. f ∈ k-U CV

⇐⇒

g ∈ k-SP ,

where g ( z) = zf  ( z) ( z ∈ U ).

It is well known that (cf. [8]) f ∈ k-U CV (respectively f ∈ k-SP ) if only if the values of p ( z) = 1 +

zf  ( z) f  ( z)



respectively

zf  ( z) f ( z)



(z ∈ U )

*

Corresponding author. E-mail addresses: [email protected] (A.K. Mishra), [email protected] (P. Gochhayat). 1 The present investigation is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India under Grant No. 48/2/2003-R&D-II. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.06.009

© 2008 Elsevier Inc.

All rights reserved.

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lie in the conic region Ωk in the w-plane, where

  Ωk := w = u + i v ∈ C: u 2 > k2 (u − 1)2 + k2 v 2 ; u > 0, 0  k < ∞ .

(1.2)

For details of the geometric description of Ωk see [8,9]. Let

    P := p: p ∈ A, p (0) = 1 and  p (z) > 0; z ∈ U and

  PC k := p: p ∈ P and p (U ) ⊂ Ωk . If f and g are functions in A and given by the power series f ( z) =

∞ 

an zn

and

g ( z) =

n=0

∞ 

bn zn

(z ∈ U ),

n=0

then the Hadamard product (or Convolution) of f and g denoted by f ∗ g, is defined by

( f ∗ g )(z) =

∞ 

an bn zn = ( g ∗ f )( z)

(z ∈ U ).

n=0

Note that f ∗ g ∈ A. Let

Φ(a, c ; z) :=

∞  (a)n n=0

(c )n

zn+1



z ∈ U, c ∈ / N− 0 := {0} ∪ {−1, −2, −3, . . .}



where (λ)n is the Pochhammer symbol (or shifted factorial) defined in terms of the gamma function, by

(λ)n =

(λ + n) 1 (n = 0), = λ(λ + 1)(λ + 2) · · · (λ + n − 1) (n ∈ N := {1, 2, . . .}). (λ)

The Carlson–Shaffer operator [3] L(a, c ) is defined in terms of the Hadamard product by

  L(a, c ) f (z) := Φ(a, c ; z) ∗ f (z) (z ∈ U , f ∈ A).

(1.3)

With a view to define the Owa–Srivastava operator we need the following definition of fractional derivative. Definition 1. (Cf. [13,14], see also [19,20].) Let the function f be analytic in a simply connected region of the z-plane containing the origin. The fractional derivative of f of order λ is defined by



 D z f ( z) = λ

1

d

(1 − λ) dz

z 0

f (ζ ) dζ (z − ζ )λ

(0  λ < 1),

where the multiplicity of ( z − ζ )λ is removed by requiring log( z − ζ ) to be real when ( z − ζ ) > 0. Using Definition 1 and its known extensions involving fractional derivative and fractional integral, Owa and Srivastava [14] introduced the fractional differintegral operator Ωzλ : A0 −→ A0 defined by

 λ    Ωz f (z) = (2 − λ)zλ D λz f (z) (λ = 2, 3, . . . ; z ∈ U ).

Note that Ωz0 f ( z) = f ( z), Ωz1 f ( z) = zf  ( z) and

  λ   Ωz f (z) = L(2, 2 − λ) f (z) (0  λ < 1; z ∈ U ).

We now introduce the following class of functions. Definition 2. (See [12].) A function f ∈ A0 is said to be in the class k-SP λ (0  λ < 1, 0  k < ∞), if Ωzλ f ∈ k-SP . Or equivalently

     z(Ωzλ f ) (z)  z(Ωzλ f ) ( z)  > k − 1 (z ∈ U ).  (Ωzλ f )(z) (Ωzλ f )(z)

(1.4)

A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

565

We note that the class k-SP λ unifies many classical and recently studied subclasses of A0 related to S . In particular for k = 0, λ = 0:

0-SP 0 := S ∗ , the class of univalent starlike functions (see [4]);

for k = 0, λ = 1:

0-SP 1 := CV , the class of univalent convex functions (see [4]);

for k = 0, λ = 0:

0-SP λ := Sλ (see [17]);

for k = 1, λ = 0:

1-SP λ := SP λ (see [16]);

for k = 1, λ = 0:

1-SP 0 := SP (see [15]);

for k = 1, λ = 1:

1-SP 1 := U CV (see [6,10]);

for k = 0, λ = 0: k-SP 0 := k-SP (see [9]);

and

for k = 0, λ = 1: k-SP 1 := k-U CV (see [8]). The authors investigated the basic properties of the general class k-SP λ , such as inclusion theorem, subordination theorem, growth theorem and class preserving transforms in [12]. Fekete and Szegö [5] obtained sharp upper bounds for |μa22 − a3 | for f ∈ S when μ is real. Thus the determination of sharp upper bounds for the nonlinear functional |μa22 − a3 | for any compact family F of functions in A0 is popularly known as the Fekete–Szegö problem for F. For different subclasses of S , the Fekete–Szegö problem has been investigated by many authors including [5,11,16–18] etc. For a brief history of the Fekete–Szegö problem see [18]. In the present paper we completely solve the Fekete–Szegö problem for the class k-SP λ (0  k < ∞, 0  λ < 1). Thus in particular the problem is solved for the class k-U CV and k-SP which remains hither to unsettled. The following definitions, notations and results shall be useful in the presentation of our results. The Jacobi elliptic integral (or normal elliptic integral) of first kind (cf. [1,2], also see [20, p. 50]) is defined by

ω

F (ω, t ) =

0

dx

(1 −

x2 )(1 − t 2 x2 )

(0 < t < 1).

(1.5)

√

The function F (1, t ) := K(t ) is called the complete elliptic integral of the first kind. Changing to the variable t  = 1 − t 2 , t ∈ (0, 1), we write K (t ) := K(t  ). It should be emphasized here that the symbol  (prime) does not stand for derivative. The following properties of K(t ) and K (t ) are well known (cf. [7]). lim K(t ) =

t →0+

π 2

lim K(t ) = ∞.

,

t →1−

Moreover the function

ν (t ) =

π K (t )  2 K(t )



t ∈ (0, 1)

strictly decreases from ∞ to 0 as t moves from 0 to 1. Therefore every positive number k can be expressed as k = cosh





ν (t )

(1.6)

for some unique t ∈ (0, 1). Lemma 1. (See [7].) Let k ∈ [0, ∞) be fixed and qk be the Riemann map of U on to Ωk , satisfying qk (0) = 1 and qk (0) > 0. If qk ( z ) = 1 + Q 1 z + Q 2 z 2 + · · · then

⎧ 2 2A ⎪ ; ⎪ ⎨ 1−k2 8 Q 1 = π2 ; ⎪ ⎪ π2 ⎩

(z ∈ U )

(1.7)

0 < k < 1, k = 1,

√ ; 4(k2 −1)K2 (t ) t (1+t )

k > 1, ⎧ 2 ( A +2) ⎪ 0 < k < 1, ⎪ ⎨ 3 Q 1; 2 Q ; k = 1, Q2 = 3 1 ⎪ ⎪ ⎩ (4K2 (t )(t 2 +√6t +1)−π 2 ) Q 1 ; k > 1, 24K2 (t ) t (1+t )

where A=

2

π

arccos k,

and K(t ) is the complete elliptic integral of first kind.

(1.8)

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A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

The closed forms of the function qk ( z) are found in [7]. Lemma 2. (See [4].) Let h ∈ P where

(z ∈ U ).

h( z) = 1 + c 1 z + c 2 z2 + · · ·

(1.9)

Then

|cn |  2 (n ∈ N),   c 2 − c 2   2 and 1

(1.10)

  2   c 2 − 1 c 2   2 − |c 1 | .  2 1 2

(1.11)

We now state our results. Theorem 1. Let the function f given by (1.1) be in the class k-SP λ (0  λ < 1; 0  k < 1). Then

⎧ 2 2 (3−λ)(2−λ) A 2 3(2−λ) A 2 μ ⎪ ( (3−λ)(1−k2 ) − (67(−1k−k)2A) − 13 ); μ  ρ1 , ⎪ 3(1−k2 ) ⎪ ⎨  2  μa − a3   (3−λ)(2−λ) A 2 ; ρ2  μ  ρ1 , 2 6(1−k2 ) ⎪ ⎪ ⎪ ⎩ (3−λ)(2−λ) A 2 ( 1 + (7−k2 ) A 2 − 3(2−λ) A 2 μ ); μ  ρ , 2 3 3(1−k2 ) 6(1−k2 ) (3−λ)(1−k2 )

(1.12)

where the constant A is given by (1.8),

ρ1 =

3−λ



5(1 − k2 )

2−λ

18 A 2

+

7 − k2 18

 and

ρ2 =

3−λ



(7 − k2 )

2−λ

18

−

1 − k2



18 A 2

.

(1.13)

Each of the estimates in (1.12) is sharp. Theorem 2. Let the function f given by (1.1) be in the class k-SP λ (0  λ < 1; 1  k < ∞) and t be the unique positive number in the open interval (0, 1) defined by (1.6). Then

⎧ (3−λ)(2−λ) Q 3Q (2−λ)μ 2 2 2 1 ( (13−λ) − Q 1 − 4K (t )(2t +√6t +1)−π ); μ  δ1 , ⎪ ⎪ 12 24K (t ) t (1+t ) ⎨  2  μa − a3   (3−λ)(2−λ) Q 1 ; δ2  μ  δ1 , 2 12 ⎪ ⎪ ⎩ (3−λ)(2−λ) Q 1 3Q 1 (2−λ)μ 4K2 (t )(t 2 +√ 6t +1)−π 2 (Q 1 + − (3−λ) ); μ  δ2 , 2 12

(1.14)

24K (t ) t (1+t )

where K(t ) is the complete elliptic integral of the first kind, Q 1 is given in (1.7),

δ1 =



3−λ 3Q 1 (2 − λ)

and

δ2 =

1 + Q1 +



3−λ 3Q 1 (2 − λ)

Q1 − 1 +

4K2 (t )(t 2 + 6t + 1) − π 2



√ 24K2 (t ) t (1 + t )

4K2 (t )(t 2 + 6t + 1) − π 2

(1.15)



√ 24K2 (t ) t (1 + t )

(1.16)

.

Each of the estimates in (1.14) is sharp. Taking λ → 1− and λ = 0 in our Theorems 1 and 2, the Fekete–Szegö problem is completely solved for the classes k-U CV and k-SP respectively. Corollary 1. Let the function f given by (1.1) be in the class k-U CV (0  k < 1). Then

⎧ 2 2 2 2 A2 ⎪ ( 3 A μ − (67(−1k−k)2A) − 13 ); μ  ρ1∗ , ⎪ ⎪ 3(1−k2 ) 2(1−k2 ) ⎨  2  A2 μa − a3   ; ρ2∗  μ  ρ1∗ , 2 3(1−k2 ) ⎪ ⎪ 2 2 2 ⎪ 2 ⎩ 2 A ( 1 + (7−k ) A − 3 A μ ); μ  ρ ∗ , 2 3(1−k2 ) 3 6(1−k2 ) 2(1−k2 )

where the constant A is given by (1.8),

ρ1∗ =

5(1 − k2 ) 9 A2

+

7 − k2 9

and

Each of the estimates in (1.17) is sharp.

ρ2∗ =

(7 − k2 ) 9

−

1 − k2 9 A2

.

(1.17)

A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

567

Corollary 2. Let the function f given by (1.1) be in the class k-U CV (1  k < ∞) and t be the unique positive number in the open interval (0, 1) defined by (1.6). Then

 2  μa − a3   2

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2 2 2 Q 1 3Q 1 μ ( 2 − Q 1 − 4K (t )(2t +√6t +1)−π ); 6 24K (t ) t (1+t ) Q1 ; 6 2 2 2 Q1 ( Q 1 + 4K (t )(2t +√6t +1)−π − 3Q21 μ ); 6 24K (t ) t (1+t )

μ  δ1∗ , δ2∗  μ  δ1∗ ,

(1.18)

μ  δ2∗ ,

where K(t ) is the complete elliptic integral of the first kind, Q 1 is given in (1.7),

δ1∗ =



2 3Q 1

and



2

∗

δ2 =

1 + Q1 +

Q1 − 1 +

3Q 1

4K2 (t )(t 2 + 6t + 1) − π 2



√ 24K2 (t ) t (1 + t )

4K2 (t )(t 2 + 6t + 1) − π 2



√

.

24K2 (t ) t (1 + t )

Each of the estimates in (1.18) is sharp. Corollary 3. Let the function f given by (1.1) be in the class k-SP (0  k < 1). Then

⎧ 2 2 2 2 A2 ⎪ ( 2 A μ − (67(−1k−k)2A) − 13 ); μ  ρ1∗∗ , ⎪ ⎪ (1−k2 ) (1−k2 ) ⎨  2  A2 μa − a3   ; ρ2∗∗  μ  ρ1∗∗ , 2 (1−k2 ) ⎪ ⎪ 2 ⎪ 2 A 2 1 (7−k2 ) A 2 ⎩ ( + 6(1−k2 ) − (21A−kμ2 ) ); μ  ρ2∗∗ , (1−k2 ) 3

(1.19)

where the constant A is given by (1.8),

ρ1∗∗ =

5(1 − k2 ) 12 A 2

+

7 − k2

and

12

ρ2∗∗ =

(7 − k2 ) 12

−

1 − k2 12 A 2

.

Each of the estimates in (1.19) is sharp. Corollary 4. Let the function f given by (1.1) be in the class k-SP (1  k < ∞) and t be the unique positive number in the open interval (0, 1) defined by (1.6). Then

 2  μa − a3   2

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2 2 2 Q1 (2Q 1 − Q 1 − 4K (t )(2t +√6t +1)−π ); 2 24K (t ) t (1+t ) Q1 ; 2 2 2 2 Q1 ( Q 1 + 4K (t )(2t +√6t +1)−π − 2Q 1 ); 2 24K (t ) t (1+t )

μ

μ

μ  δ1∗∗ , δ2∗∗  μ  δ1∗∗ ,

(1.20)

μ  δ2∗∗ ,

where K(t ) is the complete elliptic integral of the first kind, Q 1 is given in (1.7),

δ1∗∗ =

1

1 + Q1 +

2Q 1

and

δ2∗∗ =



1



2Q 1

Q1 − 1 +

4K2 (t )(t 2 + 6t + 1) − π 2



√ 24K2 (t ) t (1 + t )

4K2 (t )(t 2 + 6t + 1) − π 2

√ 24K2 (t ) t (1 + t )

 .

Each of the estimates in (1.20) is sharp. The second part of the assertion (1.12) can be improved as Theorem 3. Let the function f given by (1.1) be in the class k-SP λ (0  λ < 1; 0  k < 1). Then

 2 2   2  2  a2   (3 − λ)(2 − λ) A μa − a3  + μ − 3 − λ (7 − k ) − 1 − k 2 2 2 2 2−λ 18 18 A 6(1 − k )

and

  2 2 2  2    μa − a3  + 3 − λ 5(1 − k ) + 7 − k − μ a2   (3 − λ)(2 − λ) A 2 2 2 2 2−λ 18 18 A 6(1 − k )

(ρ2  μ  ρ3 )

(ρ3  μ  ρ1 )

(1.21)

(1.22)

where ρ1 and ρ2 are given by (1.13), and

ρ3 =

3−λ 2−λ



7 − k2 18

+

1 − k2 9 A2



.

(1.23)

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A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

The second part of the assertion (1.14) can be improved as Theorem 4. Let the function f given by (1.1) be in the class k-SP λ (0  λ < 1; 1  k < ∞) and t be the unique positive number in the open interval (0, 1) defined by (1.6). Then

 2  μa − a3  + μ − 2 

3Q 1 (2 − λ)

(3 − λ)(2 − λ) Q 1

 2  μa − a3  +



Q1 − 1 +

4K2 (t )(t 2 + 6t + 1) − π 2



√ 24K2 (t ) t (1 + t )

 2 a  2

(δ2  μ  δ3 )

12

and



3−λ

(1.24)



3−λ

4K2 (t )(t 2 + 6t + 1) − π 2

1 + Q1 +

2



√ 24K2 (t ) t (1 + t )

3Q 1 (2 − λ) (3 − λ)(2 − λ) Q 1 (δ3  μ  δ1 ) 

   − μ a22  (1.25)

12

where δ1 and δ2 are given as before, by (1.15) and (1.16) respectively, and

δ3 =

  (3 − λ) 4K2 (t )(t 2 + 6t + 1) − π 2 . Q1 + √ 3(2 − λ) Q 1 24K2 (t ) t (1 + t )

(1.26)

Finally we introduce the following functions which will be used in the discussion of sharpness of our results. Define the function G on U by

G ( z) =

1





 z

L(2 − λ, 2) z exp

z

qk (ζ ) − 1

ζ

 (z ∈ U ),

dζ

(1.27)

0

and write the function ψ( z, θ, η) in k-SP λ by

 z   ψ(z, θ, η) = L(2 − λ, 2)z exp

qk

e i θ ζ (ζ + η) 1 + ηζ

0





−1

dζ



ζ

(0  θ  2π ; 0  η  1).

(1.28)

Note that ψ( z, 0, 1) = zG ( z) defined by (1.27) and ψ( z, θ, 0) is an odd function. 2. Proofs of the theorems The following calculations shall be used in the proofs of each of Theorems 1, 2, 3 and 4. By Definition 2 there exists a function w ∈ A satisfying the conditions of the Schwarz lemma such that z(Ωzλ f ) ( z)

(Ωzλ f )(z)

  = qk w (z) (z ∈ U ),

(2.1)

where qk is the function defined as in Lemma 1. Let the function p 1 in P be defined by p 1 ( z) =

1 + w ( z)

= 1 + c 1 z + c 2 z2 + · · ·

1 − w ( z)

(z ∈ U ).

This gives w ( z) =

c1 2

z+

1 2

 c2 −

c 12

 z2 + · · ·

2

and







qk w ( z ) = 1 + Q 1

=1+

c1 2

Q 1 c1 2

z+

1 2

 c2 −



z+

1 2

c2 −

c 12 2





2

c1 2



z2 + · · · + Q 2 Q1 +

1 4



c1 2

z+

c 12 Q 2 z2 + · · · .

1 2

 c2 −

c 12 2



2 z2 + · · ·

+ ··· (2.2)

Using (2.2) in (2.1) we find that a2 =

(3 − λ) Q 1 c 1 (3)(2 − λ) 2

(2.3)

A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

and a3 =

    c2 c2 c2 Q1 (4 − λ) Q 12 1 + c2 − 1 + Q 2 1 . 2(4)(2 − λ) 4 2 2 4

569

(2.4)

Proof of Theorem 1. Putting the values of Q 1 and Q 2 for 0  k < 1 from Lemma 1 in (2.3) and (2.4) we get

(3 − λ) A 2 c1 , (3)(2 − λ)(1 − k2 )    1 (4 − λ) A 2 (7 − k2 ) A 2 2 a3 = c2 − 1− c1 , 6 2(4)(2 − λ)(1 − k2 ) (1 − k2 ) a2 =

and

μa22 − a3 =

(3 − λ)(2 − λ) A 2 24(1 − k2 )



6(2 − λ) A 2 μ

(3 − λ)(1 − k2 )

+

1

 1−

3

(7 − k2 ) A 2 (1 − k2 )





c 12 − 2c 2 .

(2.5)

Therefore

  2  2 2 2    2    μa − a3   (3 − λ)(2 − λ) A  6(2 − λ) A μ − 5 − (7 − k ) A c 2  + 2c 2 − c 2  . 2 1  (3 − λ)(1 − k2 ) 3 24(1 − k2 ) 3(1 − k2 )  1

(2.6)

If μ  ρ1 , the expression inside the first modulus symbol on the right-hand side of (2.6) is nonnegative. Thus applying Lemma 2 we get

  (7 − k2 ) A 2 4 + 4 (3 − λ)(1 − k2 ) 3 3(1 − k2 )  2 2 (3 − λ)(2 − λ) A (7 − k2 ) A 2 3(2 − λ) A μ 1 = − − 3(1 − k2 ) (3 − λ)(1 − k2 ) 3 6(1 − k2 )

2  2  μa − a3   (3 − λ)(2 − λ) A 2 24(1 − k2 )



6(2 − λ) A 2 μ

−

5

−

(2.7)

which is the first assertion of (1.12). Next, if μ  ρ2 then we write (2.5) as

μa22 − a3 =

(3 − λ)(2 − λ) A 2 24(1 − k2 )



  6(2 − λ) A 2 μ (7 − k2 ) A 2 1 2 − − c + 2c 2 . 3 3(1 − k2 ) (3 − λ)(1 − k2 ) 1

(2.8)

Applying Lemma 2 we get

  (7 − k2 ) A 2 1 6(2 − λ) A 2 μ − 4 + 4 − 3 3(1 − k2 ) (3 − λ)(1 − k2 )  2 2 2 1 3(2 − λ) A 2 μ (3 − λ)(2 − λ) A (7 − k ) A − = + 3 3(1 − k2 ) 6(1 − k2 ) (3 − λ)(1 − k2 )

2  2  μa − a3   (3 − λ)(2 − λ) A 2 24(1 − k2 )



which is the third assertion of (1.12). Lastly from (2.8) we have

    2 2 2 2  2   μa − a3  = (3 − λ)(2 − λ) A 2c 2 − c 2 + (7 − k ) A + 2 − 6(2 − λ) A μ c 2  2 1 1  2 2 2 3 24(1 − k ) 3(1 − k ) (3 − λ)(1 − k )      1 2   (7 − k2 ) A 2 2 6(2 − λ) A 2 μ  2  (3 − λ)(2 − λ) A 2  c  . + c 2 c − −  + 2 1 1  2   3(1 − k2 ) 3 24(1 − k2 ) (3 − λ)(1 − k2 ) 

(2.9)

ρ2 < μ < ρ1 implies    (7 − k2 ) A 2 2 6(2 − λ) A 2 μ    1.  − +  3(1 − k2 ) 3 (3 − λ)(1 − k2 ) 

We observe that

Thus applying Lemma 2 to (2.9) we get 2  2  μa − a3   (3 − λ)(2 − λ) A 2 2 6(1 − k )

(2.10)

which is the second part of the assertion (1.12). We next discuss sharpness of the estimates in (1.12). If μ > ρ1 , equality holds in (1.12) if and only if equality holds in (2.7). This happens if and only if |c 1 | = 2 and |c 12 − c 2 | = 2. Thus w (z) = z. Equivalently the extremal function is of the form ψ(z, 0, 1) defined by (1.28) or one of its rotations.

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A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

If μ < ρ2 then equality holds in (2.10) if and only if c 1 = 0 and |c 2 | = 2. Thus w ( z) = e i θ z2 and the extremal function is ψ(z, θ, 0) or one of its rotations. If μ = ρ2 , the equality holds if and only if |c 2 | = 2. Equivalently, we have     1+η 1+z 1−η 1−z p 1 ( z) = + (0 < η < 1; z ∈ U ). 2 1−z 2 1+z Thus the extremal function f is ψ( z, 0, η) or one of its rotations. Similarly, μ = ρ1 is equivalent to 6(2 − λ) A 2 μ

(3 − λ)(1 − k2 )

−

5 3

−

(7 − k2 ) A 2 = 0. 3(1 − k2 )

Therefore equality holds true in (2.7) if and only if |c 12 − c 2 | = 2. This happens if and only if 1 p 1 ( z)

=

1+η



1+z



+

1−z

2

1−η



1+z



(0 < η < 1; z ∈ U ).

1−z

2

Thus the extremal function is ψ( z, π , η) or one of its rotations. Lastly if ρ2 < μ < ρ1 , then equality holds true if |c 1 | = 0 and |c 2 | = 2. Equivalently we have p 1 ( z) =

1 + η z2 1 − η z2

(0  η  1; z ∈ U ).

Thus the function f is ψ( z, 0, 0) or one of its rotation. The proof of Theorem 1 is complete.

2

Proof of Theorem 2. For k = 1 the result is due to Srivastava and Mishra [16]. Therefore we prove the theorem for k > 1. We follow the lines of proof of Theorem 1 and give here only those steps where we differ. Putting the values of Q 2 for k > 1 (from Lemma 1) in (2.4) we get

    2 2  2    6t + 1) − π 2 2 μa − a3  = (3 − λ)(2 − λ) Q 1  3(2 − λ) Q 1 μ − Q 1 − 1 + 4K (t )(t +√  c − 2c 2 2 1   2 48 (3 − λ) 24K (t ) t (1 + t )    2 2 2  2  4K (t )(t + 6t + 1) − π  2  (3 − λ)(2 − λ) Q 1  3(2 − λ) Q 1 μ      −  1 − Q − c 2 c − c + √ 1 2 . 1  (3 − λ)  1 48 24K2 (t ) t (1 + t )

(2.11) (2.12)

If μ  δ1 , then the expression inside the first modulus symbol on the right-hand side of Eq. (2.12) is nonnegative. Thus (1.10) and (1.11) yield

 2  μa − a3   (3 − λ)(2 − λ) Q 1 2



3(2 − λ) Q 1 μ

(3 − λ)

12

− Q1 −

4K2 (t )(t 2 + 6t + 1) − π 2



√ 24K2 (t ) t (1 + t )

(2.13)

which is the first part of assertion (1.14). If μ  δ2 then Q1 − 1 +

4K2 (t )(t 2 + 6t + 1) − π 2

√

24K2 (t ) t (1 + t )

−

3(2 − λ) Q 1 μ

(3 − λ)

 0.

Therefore, applying Lemma 2 in (2.11) we get

 2  μa − a3  = (3 − λ)(2 − λ) Q 1 2

48

=

(3 − λ)(2 − λ) Q 1 12

 Q1 − 1 +

Q1 +

4K2 (t )(t 2 + 6t + 1) − π 2

√

24K2 (t ) t (1 + t )

4K2 (t )(t 2 + 6t + 1) − π 2

√ 24K2 (t ) t (1 + t )

−



3(2 − λ) Q 1 μ  2  c  + 2|c 2 | − 1 (3 − λ)

3(2 − λ) Q 1 μ

(3 − λ)





.

This is precisely the third part of our assertion (1.14). Lastly, if δ2 < μ < δ1 then

  2 2  6t + 1) − π 2 3(2 − λ) Q 1 μ   Q 1 + 4K (t )(t +√   1. −  (3 − λ)  24K2 (t ) t (1 + t )

Again applying Lemma 2 in (2.11) we have

   2 2   2  6t + 1) − π 2 3Q 1 (2 − λ)μ 2  μa − a3  = (3 − λ)(2 − λ) Q 1 2c 2 − c 2 + Q 1 − 1 + 1 + 4K (t )(t +√ − c1  2 1  48 (3 − λ) 24K2 (t ) t (1 + t )    (3 − λ)(2 − λ) Q 1  1     2c 2 − c 12  + c 12  48

2

(2.14)

A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

 =

(3 − λ)(2 − λ) Q 1

2 2−

48

(3 − λ)(2 − λ) Q 1 12





  1  2 c  + c 2  1 2 1

(2.15)

.

Thus we get the second part of the assertion (1.14). The sharpness of the estimates in (1.14) can be demonstrated as in Theorem 1. Proof of Theorem 3. Suppose 0 < k < 1 and

2

ρ2  μ  ρ3 . Using (2.9) for |μa22 − a3 | and (2.3) for |a2 | we have

 2 2     2   μa − a3  + (μ − ρ2 )|a2 |2 = μa2 − a3  + μ − 3 − λ (7 − k ) − 1 − k a2  2 2 2 2−λ 18 18 A 2      (3 − λ)(2 − λ) A 2  1 2   (7 − k2 ) A 2 2 6(2 − λ) A 2 μ  2  c  + c + 2 c − −  2  2 1   3(1 − k2 ) 3 24(1 − k2 ) (3 − λ)(1 − k2 )  1 2     3 − λ (7 − k2 ) 1 − k2 (3 − λ) A 2   + μ− − c 1  2 2 2−λ 18 18 A (3)(2 − λ)(1 − k )     1 2   (7 − k2 ) A 2 2 6(2 − λ) A 2 μ  2  (3 − λ)(2 − λ) A 2  c    c = 2 c − − + +  2 2 1   3(1 − k2 ) 3 24(1 − k2 ) (3 − λ)(1 − k2 )  1    (7 − k2 ) A 2 1  2  6(2 − λ) A 2 μ − + c + 1 . 3 (3 − λ)(1 − k2 ) 3(1 − k2 ) Observe that, since

μ  ρ3

(7 − k2 ) A 2 2 6(2 − λ) A 2 μ + −  0. 2 3 3(1 − k ) (3 − λ)(1 − k2 ) Therefore

  2   2      μa − a3   (3 − λ)(2 − λ) A 2 2 − 1 c 2  + c 2  . 2 1 2 1 24(1 − k2 ) An application of Lemma 2 gives 2      2  μa − a3   (3 − λ)(2 − λ) A 4 − c 2  + c 2  2 1 1 2 24(1 − k )

=

(3 − λ)(2 − λ) A 2 6(1 − k2 )

(ρ2  μ  ρ3 ).

This establishes (1.21). Similarly, for the value of

μ prescribed in (1.22), we have

  2 2    2    μa − a3  + (ρ1 − μ)|a2 |2 = μa2 − a3  + 3 − λ 5(1 − k ) + 7 − k − μ a2  2 2 2 2 2−λ 18 18 A        1 2   (7 − k2 ) A 2 2 6(2 − λ) A 2 μ  2  (3 − λ)(2 − λ) A 2       c 2 c − − c  + +  2 2 1   3(1 − k2 ) 3 24(1 − k2 ) (3 − λ)(1 − k2 )  1    (2 − λ)2 A 4  2  3 − λ 5(1 − k2 ) 7 − k2 −μ + + c 2 2−λ 18 18 A 4(1 − k2 )2 1      (7 − k2 ) A 2 2  2  1 2   6(2 − λ) A 2 μ (3 − λ)(2 − λ) A 2  + − − c 2 c − c = 2  2 1   (3 − λ)(1 − k2 ) 3  1 24(1 − k2 ) 3(1 − k2 )    (3 − λ)(2 − λ) A 2 6 A 2 5(1 − k2 ) 7 − k2 6(2 − λ) A 2 μ  2  c . + + − 1 2 2 2 18 24(1 − k ) 1−k 18 A (3 − λ)(1 − k2 )

Since

μ  ρ3 , 6(2 − λ) A 2 μ

(3 − λ)(1 − k2 ) and we get

−

571



(7 − k2 ) A 2 2 − 0 3 3(1 − k2 )

572

A.K. Mishra, P. Gochhayat / J. Math. Anal. Appl. 347 (2008) 563–572

   2       2  μa − a3   (3 − λ)(2 − λ) A 2 c 2 − 1 c 2  + c 2  2 1 1   2 24(1 − k2 ) =

(3 − λ)(2 − λ) A 2 6(1 − k2 )

which proves (1.22). The proof of Theorem 3 is complete.

2

The proof of Theorem 4 is similar to that of Theorem 3, except for obvious changes. Therefore, we omit the details. 3. Conclusion Remark 1. Results on Fekete–Szegö problem found in [11,16,17] are included in Theorem 1 for the particular cases k → 1− , λ → 1− ; k → 1− and k = 0 respectively. Remark 2. Since the function qk is continuous with respect to the parameter k [7], taking k → 1− in Theorem 1 and k = 1 in Theorem 2 we get identical results. This can also be verified directly using the limiting behavior of K(t ) cited in the introduction. Supplementary material The online version of this article contains additional supplementary material. Please visit DOI: 10.1016/j.jmaa.2008.06.009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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