A. K. MISHRA AND P. GOCHHAYAT KODAI MATH. J. 33 (2010), 310–328
¨ PROBLEM FOR A CLASS DEFINED FEKETE-SZEGO BY AN INTEGRAL OPERATOR Akshaya Kumar Mishra and Priyabrat Gochhayat1 Abstract By making use of an integral operator due to Noor, a new subclass of analytic functions, denoted by k UCVn ðn A N0 :¼ f0; 1; 2; . . .g; 0 a k < yÞ; is introduced. For this class the Fekete-Szego¨ problem is completely settled. The results obtained here also give the Fekete-Szego¨ inequalities for the classes of k-uniformly convex functions and k-parabolic starlike functions.
1.
Introduction and definitions
Let A denote the family of functions analytic in the open unit disk U :¼ fz : z A C; jzj < 1g and let A0 be the class of functions f in A satisfying the normalization condition f ð0Þ ¼ f 0 ð0Þ 1 ¼ 0. Thus, the functions in A0 are given by the power series y X ð1:1Þ an z n ðz A UÞ: f ðzÞ ¼ z þ n¼2 n
Let D : A0 ! A0 be the operator defined by z D n f ðzÞ ¼ f ðzÞ n > 1 ð1 zÞ nþ1 y X kþn1 ¼zþ ak z k ; ð f A A0 ; n A N0 :¼ f0; 1; 2; . . .gÞ; k 1 k¼2 0 0
denotes the convolution or Hadamard product. We note that zðz n1 f ÞðnÞ . The operator D n f is called the D 0 f ¼ f , D 1 f ¼ zf 0 and D n f ¼ n! where
2000 Mathematics Subject Classification: Primary 30C45. Keywords and Phrases: Integral operator, k-uniformly convex function, elliptic integral, FeketeSzego¨ problem, Gauss hypergeometric series. 1 (Corresponding author) The present research work of the second author was supported by Council of Scientific and Industrial Research, New Delhi, Government of India, under Senior Research Fellowship grant. Sanction letter no. 9/297/(73)/2008—EMR-I. Received October 21, 2008; revised December 8, 2009.
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311
fekete-szego¨ problem
Ruscheweyh derivative of n th order of f . Analogous to D n f , Noor [13] defined an integral operator In : A0 ! A0 as follows: Write fn ðzÞ ¼ z=ð1 zÞ nþ1 and let fny be defined by the relation fn ðzÞ fny ðzÞ ¼
ð1:2Þ
z ð1 zÞ 2
:
For f A A0 let In f ðzÞ ¼ fny ðzÞ f ðzÞ:
ð1:3Þ
Note that I0 f ðzÞ ¼ zf 0 ðzÞ and I1 f ðzÞ ¼ f ðzÞ. The operator In f defined by (1.3) is called n th order Noor integral operator of f . For details see (cf. [13], [14], also see [15]). For any real or complex numbers a, b, c other than 0; 1; 2; . . . ; the Gauss hypergeometric series is defined by ð1:4Þ
2 F1 ða; b; c; zÞ
¼1þ
ab aða þ 1Þbðb þ 1Þ z 2 zþ þ : c cðc þ 1Þ 2!
We can write (1.3) as ð1:5Þ
In f ðzÞ ¼ ½z2 F1 ð2; 1; n þ 1; zÞ f ðzÞ " # y X k! k z f ðzÞ: ¼ zþ ðn þ 1Þ ðn þ k 1Þ k¼2
Let S, S , CV, UCV and SP denote, respectively the subclasses of A0 consisting of functions which are univalent, starlike, convex (cf. [4]), uniformly convex (cf. [6]) and parabolic starlike (cf. [16])) in U. For fixed k ð0 a k < yÞ, the function f A A0 is said to be in k UCV; the class of k-uniformly convex functions in U, if the image of every circular arc g contained in U, with centre x where jxj a k, is a convex arc. This interesting unification of the concepts of convex functions and uniformly convex functions is due to [8]. The class k SP, consisting of k-parabolic starlike functions, is defined from k UCV via the Alexander’s transform (see [9]) i.e. ð1:6Þ
f A k UCV , g A k SP;
where gðzÞ ¼ zf 0 ðzÞ ðz A UÞ:
The one variable characterization theorem (cf. [8]) of the the class k UCV gives that f A k UCV (respectively f A k SP) if only if the values of zf 00 ðzÞ zf 0 ðzÞ respectively pðzÞ ¼ 1 þ 0 ðz A UÞ f ðzÞ f ðzÞ lie in the conic region Wk in the w-plane, where Wk :¼ fw ¼ u þ iv A C : u 2 > k 2 ðu 1Þ 2 þ k 2 v 2 ; 0 a k < yg:
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For details of the geometric description of Wk see [8, 9]. Taking cue from (1.6), we now define a new subclass of analytic functions by using the linear operator In . Definition 1. The function f A A0 is said to be in the class k UCVn ð0 a k < y; n A N0 Þ if In f A k SP. Or equivalently zðIn f Þ 0 ðzÞ zðIn f Þ 0 ðzÞ 1 ðz A UÞ: < > k ðIn f ÞðzÞ ðIn f ÞðzÞ Note that the class k UCVn unifies many subclasses of A0 related to S. In particular, for k ¼ 0, n ¼ 0 : 0 UCV0 :¼ CV, the class of univalent convex functions (see [4]); for k ¼ 1, n ¼ 0 : 1 UCV0 :¼ UCV, the class of uniformly convex functions (see [6]); for k ¼ 0, n ¼ 1 : 0 UCV1 :¼ S , the class of univalent starlike functions (see [4]); for k ¼ 1, n ¼ 1 : 1 UCV1 :¼ SP, the class of parabolic starlike functions (see [16]); for k 0 0, n ¼ 0 : k UCV0 :¼ k UCV (see [8]); for k 0 0, n ¼ 1 : k UCV1 :¼ k SP (see [8]). It is well known (cf. [4]) that for f A S and given by (1.1), the sharp inequality ja3 a22 j a 1 holds. Fekete-Szego¨, in their seminal work [5], found sharp bounds for jma22 a3 j, ðm A R; 0 a m < 1Þ, for f A S. Thus, for any family F of functions in A0 , the problem of finding sharp estimates on the nonlinear functional jma22 a3 j, ðm A R or m A CÞ is popularly known as the Fekete-Szego¨ problem for F. Recently the present authors [12] settled this problem for the classes k UCV, k SP and some related classes defined using fractional calculus. One may also see [7, 10, 11, 17, 18] for some interesting results on this topic. In the present paper the Fekete-Szego¨ problem for the class k UCVn ð0 a k < y; n A N0 Þ is settled completely. For k ¼ 1, n ¼ 0 and n ¼ 0; n ¼ 1, the result on Fekete-Szego¨ inequalities obtained here include, respectively, earlier results of Ma and Minda [11] on UCV and recent results of the authors on k UCV and k SP in [12]. For values of n > 1 our results provide new information. In the present investigation we also need the following definitions and notations, for the presentation of our results. The Jacobi elliptic integral (or normal elliptic integral) of first kind (cf. [1], [2], also see [19, p. 50]) is defined by ðo dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0 < t < 1Þ: Fðo; tÞ ¼ 2 ð1 x Þð1 t 2 x 2 Þ 0 The function Fð1; tÞ :¼ KðtÞ is calledpthe complete elliptic integral of the ffiffiffiffiffiffiffiffiffiffiffiffi first kind. Changing to the variable t 0 ¼ 1 t 2 , t A ð0; 1Þ, we write K 0 ðtÞ :¼ Kðt 0 Þ. It should be emphasized here that the symbol 0 (prime) does not stand for derivative. The following properties of KðtÞ and K 0 ðtÞ are well known (cf. [7]). lim KðtÞ ¼
t!0þ
p 2
lim KðtÞ ¼ y:
t!1
313
fekete-szego¨ problem Moreover the function nðtÞ ¼
p K 0 ðtÞ ; 2 KðtÞ
ðt A ð0; 1ÞÞ
strictly decreases from y to 0 as t moves from 0 to 1. Therefore every positive number k can be expressed as ð1:7Þ
k ¼ coshðnðtÞÞ
for some unique t A ð0; 1Þ. Define the function G on U by ð z qk ðzÞ 1 ð1:8Þ GðzÞ ¼ ½z2 F1 ðn þ 1; 1; 2; zÞ z exp dz ; z 0
ðz A UÞ;
where qk is the function defined in Lemma 1 (below) and write the function cðz; y; hÞ in k UCVn by ð z iy e zðz þ hÞ dz qk cðz; y; hÞ ¼ ½z2 F1 ðn þ 1; 1; 2; zÞ z exp ð1:9Þ 1 1 þ hz z 0 ð0 a y a 2p; 0 a h a 1Þ: Note that cðz; 0; 1Þ ¼ GðzÞ defined by (1.8) and cðz; y; 0Þ is an odd function. 2.
Preliminary lemmas
We need the following results in our investigation: Lemma 1 [7]. Let k A ½0; yÞ be fixed and qk be the Riemann map of U on to Wk , satisfying qk ð0Þ ¼ 1 and qk0 ð0Þ > 0. If ð2:1Þ then
qk ðzÞ ¼ 1 þ Q1 z þ Q2 z 2 þ ;
ðz A UÞ
8 > 2A 2 > > ; 0 < k < 1; > > 1 k2 > > > > > p2 > > > ; k > 1; pffiffi : 4ðk 2 1ÞK 2 ðtÞ tð1 þ tÞ 8 ðA 2 þ 2Þ > > > Q1 ; 0 < k < 1; > > 3 > > > > > > ð4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 Þ > > Q1 ; k > 1; pffiffi : 24K 2 ðtÞ tð1 þ tÞ
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akshaya kumar mishra and priyabrat gochhayat
where ð2:2Þ
A¼
2 arccos k; p
and KðtÞ is the complete elliptic integral of first kind. Lemma 2 [10]. series
Let h be analytic, with hð0Þ ¼ 1, 0 and given by the hðzÞ ¼ 1 þ c1 z þ c2 z 2 þ
ð2:3Þ
ðz A UÞ:
Then jcn j a 2
ð2:4Þ ð2:5Þ
3.
jc2 c12 j a 2 and
ðn A NÞ; 2 c2 1 c 2 a 2 jc1 j : 1 2 2
Fekete-Szego¨ inequalities
The following calculations shall be used in each of the proofs of Theorems 1, 2 and 3. By Definition 1 there exists a function w A A satisfying the conditions of the Schwarz lemma such that zðIn f ðzÞÞ 0 ¼ qk ðwðzÞÞ ðz A UÞ; ðIn f ðzÞÞ
ð3:1Þ
where qk is the function defined as in Lemma 1. Let the function p1 be defined by p1 ðzÞ ¼ so that 0.
1 þ wðzÞ ¼ 1 þ c1 z þ c2 z 2 þ ; 1 wðzÞ
ðz A UÞ;
This gives c1 1 c12 2 c2 wðzÞ ¼ z þ z þ : 2 2 2
Substituting this in the series (2.1) we get c1 1 c2 c2 1 z 2 þ zþ ð3:2Þ qk ðwðzÞÞ ¼ 1 þ Q1 2 2 2 2 c1 1 c12 2 c2 zþ þ Q2 z þ þ 2 2 2 Q 1 c1 1 c2 1 c2 1 Q1 þ c12 Q2 z 2 þ ; ¼1þ zþ 2 4 2 2
fekete-szego¨ problem
315
Using the expansion (1.5) in (3.1) and equating coe‰cients we find that ð3:3Þ
a2 ¼
ðn þ 1ÞQ1 c1 4
and ð3:4Þ
ðn þ 1Þðn þ 2Þ Q12 c12 Q1 c12 Q2 c12 þ c2 a3 ¼ þ : 12 4 2 2 4
We have the following: Theorem 1. Let the function f given by (1.1) be in the class k UCVn ð0 a k < 1Þ. Then 8 ðn þ 1Þðn þ 2ÞA 2 6ðn þ 1ÞA 2 m > > > > > ðn þ 2Þð1 k 2 Þ 6ð1 k 2 Þ > > > 2 > > ð2 þ A Þ 2A 2 > > ; m b a1 ; > > 3 ð1 k 2 Þ > > > < ðn þ 1Þðn þ 2ÞA 2 ð3:5Þ jma22 a3 j a ; a2 a m a a1 ; > 6ð1 k 2 Þ > > > > > ðn þ 1Þðn þ 2ÞA 2 2A 2 > > > > 6ð1 k 2 Þ ð1 k 2 Þ > > > > 2 > ð2 þ A Þ 6ðn þ 1ÞA 2 m > > : þ ; m a a2 ; 3 ðn þ 2Þð1 k 2 Þ where the constant A is given by (2.2), ð3:6Þ and
ðn þ 2Þ ð5 þ A 2 Þð1 k 2 Þ 1 a1 :¼ a1 ðkÞ ¼ þ ðn þ 1Þ 18A 2 3 ðn þ 2Þ 1 ð1 A 2 Þð1 k 2 Þ : a2 :¼ a2 ðkÞ ¼ ðn þ 1Þ 3 18A 2
Each of the estimates in (3.5) is sharp. Proof. Putting the value of Q1 and Q2 for 0 a k < 1 from Lemma 1 in (3.3) and (3.4) we get a2 ¼
ðn þ 1ÞA 2 c1 2ð1 k 2 Þ
and a3 ¼
ðn þ 1Þðn þ 2ÞA 2 1 6A 2 2 ð1 A c Þ c2 : 2 6 12ð1 k 2 Þ 1 k2 1
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akshaya kumar mishra and priyabrat gochhayat
Therefore
ð3:7Þ
ð3:8Þ
ðn þ 1Þ 2 A 4 c 2 ðn þ 1Þðn þ 2ÞA 2 2 1 jma2 a3 j ¼ m c2 4ð1 k 2 Þ 2 12ð1 k 2 Þ 2 1 6A ð1 A 2 Þ c2 6 1 k2 1 ( ðn þ 1Þ 2 A 4 ðn þ 1Þðn þ 2ÞA 2 ¼ mþ ð1 A 2 Þ 4ð1 k 2 Þ 2 72ð1 k 2 Þ ) 2 6A 2 ðn þ 1Þðn þ 2ÞA 2 c c 2 1 2 2 ð1 k Þ 12ð1 k Þ ðn þ 1Þðn þ 2ÞA 2 6ðn þ 1ÞA 2 m ¼ 2 ðn þ 2Þð1 k 2 Þ 24ð1 k Þ 1 6A 2 2 2 ð1 A Þ þ c1 2c2 2 3 ð1 k Þ 2 ðn þ 1Þðn þ 2ÞA 6ðn þ 1ÞA 2 m ¼ 24ð1 k 2 Þ ðn þ 2Þð1 k 2 Þ 1 A2 2A 2 2 2 2 c þ 2c 2c þ 2 1 1 2 3 ð1 k Þ ðn þ 1Þðn þ 2ÞA 2 6ðn þ 1ÞA 2 m a ðn þ 2Þð1 k 2 Þ 24ð1 k 2 Þ 5 þ A2 2A 2 2 2 jc j þ 2jc1 c2 j : 3 ð1 k 2 Þ 1
Now we note that ð3:9Þ
6ðn þ 1ÞA 2 m 5 þ A2 2A 2 b0 ðn þ 2Þð1 k 2 Þ 3 ð1 k 2 Þ ðn þ 2Þ ð5 þ A 2 Þð1 k 2 Þ 1 provided m b þ ¼ a1 : ðn þ 1Þ 18A 2 3
Thus if m b a1 , the expression inside the first modulus symbol on the right hand side of (3.8) is non negative. An application of Lemma 2 yields, ðn þ 1Þðn þ 2ÞA 2 6ðn þ 1ÞA 2 m 2 jma2 a3 j a ð3:10Þ ðn þ 2Þð1 k 2 Þ 24ð1 k 2 Þ 5 þ A2 2A 2 4 þ 4 3 ð1 k 2 Þ
fekete-szego¨ problem ðn þ 1Þðn þ 2ÞA 2 6ðn þ 1ÞA 2 m a ðn þ 2Þð1 k 2 Þ 6ð1 k 2 Þ 2 þ A2 2A 2 : 3 ð1 k 2 Þ which is the first part of assertion (3.5). Rewriting (3.7) ðn þ 1Þðn þ 2ÞA 2 2A 2 1 A2 2 ð3:11Þ 2c jma2 a3 j ¼ 2 24ð1 k 2 Þ ð1 k 2 Þ 3 2 6ðn þ 1ÞA m c2 ðn þ 2Þð1 k 2 Þ 1 ðn þ 1Þðn þ 2ÞA 2 2A 2 1 A2 ¼ 2 2 24ð1 k Þ ð1 k Þ 3 2 6ðn þ 1ÞA m c12 þ 2c2 : 2 ðn þ 2Þð1 k Þ Now if m a a2 , where a2 is given by (3.6), then ðn þ 1Þðn þ 2ÞA 2 2A 2 1 A2 2 jma2 a3 j a 24ð1 k 2 Þ ð1 k 2 Þ 3 2 6ðn þ 1ÞA m jc12 j þ 2jc2 j ðn þ 2Þð1 k 2 Þ Applying Lemma 2, we get ð3:12Þ
ðn þ 1Þðn þ 2ÞA 2 24ð1 k 2 Þ
2A 2 1 A2 2 ð1 k Þ 3 2 6ðn þ 1ÞA m 4þ4 ðn þ 2Þð1 k 2 Þ ðn þ 1Þðn þ 2ÞA 2 2A 2 2 þ A2 6ðn þ 1ÞA 2 m þ ¼ ðn þ 2Þð1 k 2 Þ 6ð1 k 2 Þ ð1 k 2 Þ 3
jma22 a3 j a
which is the third part of the assertion (3.5). Lastly from (3.11), write ðn þ 1Þðn þ 2ÞA 2 2A 2 1 A2 ð3:13Þ 2c jma22 a3 j ¼ þ 2 2 2 24ð1 k Þ ð1 k Þ 3 6ðn þ 1ÞA 2 m c12 2 ðn þ 2Þð1 k Þ
317
318
akshaya kumar mishra and priyabrat gochhayat ðn þ 1Þðn þ 2ÞA 2 1 2 a 2c2 c1 2 24ð1 k 2 Þ 2A 2 2 þ A2 6ðn þ 1ÞA 2 m 2 þ þ jc j : ðn þ 2Þð1 k 2 Þ 1 ð1 k 2 Þ 3
Observe that a2 < m < a1 gives 2A 2 2 þ A2 6ðn þ 1ÞA 2 m ð3:14Þ ð1 k 2 Þ þ 3 ðn þ 2Þð1 k 2 Þ a 1: Therefore an application of Lemma 2 in (3.13) gives ð3:15Þ
jma22 a3 j a
ðn þ 1Þðn þ 2ÞA 2 6ð1 k 2 Þ
which is the second part of assertion (3.5). We next discuss the sharpness of (3.5). If m > a1 , equality holds in (3.5) if and only if equality holds in (3.10). This happens if and only if jc1 j ¼ 2 and jc12 c2 j ¼ 2. Thus wðzÞ ¼ z. Equivalently the extremal function is GðzÞ defined by (1.8) or one of its rotations. If m < a2 then equality holds in (3.12) if and only if c12 ¼ 4 and c2 ¼ 2 in (3.7) if and only if c1 ¼ 2e ip=2 or c1 ¼ 2e i3p=2 which also gives c2 ¼ 2: Thus p 3p and the extremal function is cðz; y; 1Þ or one wðzÞ ¼ e iy z where y ¼ or y ¼ 2 2 of its rotations. If m ¼ a2 , the equality holds if and only if jc2 j ¼ 2. Equivalently, we have 1þh 1þz 1h 1z p1 ðzÞ ¼ þ ð0 < h < 1; z A UÞ: 2 1z 2 1þz Thus the extremal function f is cðz; 0; hÞ or one of its rotations. Similarly if m ¼ a1 , is equivalent to 6ðn þ 1ÞA 2 m 5 þ A2 2A 2 ¼ 0: 2 ðn þ 2Þð1 k Þ 3 ð1 k 2 Þ Therefore equality holds true in (3.10) if and only if jc12 c2 j ¼ 2 in (3.8). happens if and only if 1 1þh 1þz 1h 1z ¼ þ ð0 < h < 1; z A UÞ: p1 ðzÞ 2 1z 2 1þz
This
Thus the extremal function is cðz; p; hÞ or one of its rotations. Lastly if a2 < m < a1 , then equality holds true if jc1 j ¼ 0 and jc2 j ¼ 2. Equivalently we have 1 þ hz 2 ð0 a h a 1; z A UÞ: p1 ðzÞ ¼ 1 hz 2 Thus the function f is cðz; 0; 0Þ or one of its rotation. is complete.
The proof of Theorem 1 9
fekete-szego¨ problem
319
Taking n ¼ 0 and n ¼ 1 in Theorem 1, we get the Fekete-Szego¨ inequalities for the classes k UCV and k SP ð0 a k < 1Þ respectively, obtained earlier by the authors [12]. Corollary 1. Let the function f given by (1.1) be in the class k UCV ð0 a k < 1Þ. Then 8 > 2A 2 3A 2 m ð7 k 2 ÞA 2 1 > > ; m b a1 ; > > > 3 3ð1 k 2 Þ 2ð1 k 2 Þ 6ð1 k 2 Þ > > < 2 A jma22 a3 j a ; a2 a m a a1 ; > 3ð1 k 2Þ > > > > > 2A 2 1 ð7 k 2 ÞA 2 3A 2 m > > þ ; m a a2 ; : 2ð1 k 2 Þ 3ð1 k 2 Þ 3 6ð1 k 2 Þ where the constant A is given by (2.2), a1 ¼
5ð1 k 2 Þ 7 k 2 þ 9A 2 9
and
a2 ¼
7 k2 1 k2 : 9 9A 2
The result is best possible. Corollary 2. Let the function f given by (1.1) be in the class k SP ð0 a k < 1Þ. Then 8 > 2A 2 2A 2 m ð7 k 2 ÞA 2 1 > > ; m b a > 1 ; > > 3 ð1 k 2 Þ ð1 k 2 Þ 6ð1 k 2 Þ > > < A2 jma22 a3 j a ; a 2 a m a a1 ; > ð1 k 2 Þ > > > > > 2A 2 1 ð7 k 2 ÞA 2 2A 2 m > > þ ; m a a : 2 ; ð1 k 2 Þ ð1 k 2 Þ 3 6ð1 k 2 Þ where the constant A is given by (2.2), a 1 ¼
5ð1 k 2 Þ 7 k 2 þ 12A 2 12
and
a 2 ¼
ð7 k 2 Þ 1 k 2 : 12 12A 2
The result is best possible. Theorem 2. Let the function f given by (1.1) be in the class k UCVn ðk ¼ 1; n A NÞ. Then 8 4ðn þ 1Þðn þ 2Þ 12ðn þ 1Þm 1 4 > > > ; m b b1 ; > > 3p 2 ðn þ 2Þp 2 3 p2 > > < 2ðn þ 1Þðn þ 2Þ ð3:16Þ jma22 a3 j a ; b2 a m a b1 ; > 3p 2 > > > 4ðn þ 1Þðn þ 2Þ 4 1 12ðn þ 1Þm > > > þ ; m a b2 ; : 3p 2 p 2 3 ðn þ 2Þp 2
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akshaya kumar mishra and priyabrat gochhayat
where,
ðn þ 2Þ 5p 2 1 b1 :¼ b 1 ðkÞ ¼ þ ðn þ 1Þ 72 3
and
ðn þ 2Þ 1 p 2 b2 :¼ b 2 ðkÞ ¼ : ðn þ 1Þ 3 72
Each of the estimates in (3.16) is sharp. Proof. We put the values of Q1 and Q2 for k ¼ 1 given in (3.5) and (3.6). By following the lines of proof of Theorem 1 the result follows. 9 Taking n ¼ 0 in Theorem 2, we get the Fekete-Szego¨ inequalities for the class UCV obtained earlier by Ma and Minda [11]. Similarly the choice n ¼ 1 yields a result for the class SP obtained recently by the present authors [12], which is as follows: Corollary 3.
Let the function f given by (1.1) be in the class SP. 8 > 8 8m 1 4 > > ; m b b1 ; > 2 p2 2 > p 3 p > > < 4 2 jma2 a3 j a ; b2 a m a b 1 ; 2 > p > > > > 8 4 1 8m > > þ ; m a b2 ; : 2 p p2 3 p2
Then
where, b1
3 5p 2 1 þ ¼ 2 72 3
and
b2
3 1 p2 ¼ : 2 3 72
The result is best possible. Theorem 3. Let the function f given by (1.1) be in the class k UCVn ðk > 1Þ. Then
ð3:17Þ
8 > > > ðn þ 1Þðn þ 2ÞQ1 3ðn þ 1Þm Q1 Q1 > > > ðn þ 2Þ 12 > > > ! > > 2 2 2 > 4K ðtÞðt þ 6t þ 1Þ p > > > ; m b g1 ; pffiffi > > 24K 2 ðtÞ tð1 þ tÞ > > > < ðn þ 1Þðn þ 2ÞQ1 2 jma2 a3 j a ; g2 a m a g1 ; > 12 > > > > > ðn þ 1Þðn þ 2ÞQ1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 > > þ Q pffiffi > 1 > > 12 24K 2 ðtÞ tð1 þ tÞ > > > ! > > > 3ðn þ 1Þm > > > Q 1 ; m a g2 ; : ðn þ 2Þ
321
fekete-szego¨ problem
where KðtÞ is the complete elliptic integral of the first kind, Q1 is given in (2.1), ! ðn þ 2Þ 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 1 þ Q1 þ g1 :¼ g1 ðkÞ ¼ pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ and ! ðn þ 2Þ 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 Q1 1 þ g2 :¼ g2 ðkÞ ¼ : pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ Each of the estimates in (3.17) is sharp. Proof. We follow the lines of proof of Theorem 1 and give here only the essential steps. Putting the value of Q2 for k > 1 from Lemma 1 in (3.4) we get Q1 ðn þ 1Þ c1 ; 4 " # Q1 ðn þ 1Þðn þ 2Þ Q1 c12 1 c12 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 c12 c2 þ a3 ¼ þ : pffiffi 12 2 4 2 4 24K 2 ðtÞ tð1 þ tÞ Therefore ( ðn þ 1Þðn þ 2ÞQ1 3Q1 ðn þ 1Þm 2 ð3:18Þ jma2 a3 j ¼ ðn þ 2Þ 48 !) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 2 Q1 1 þ 2c c p ffiffi 2 1 24K 2 ðtÞ tð1 þ tÞ ( ðn þ 1Þðn þ 2ÞQ1 3Q1 ðn þ 1Þm a 1 Q1 ð3:19Þ ðn þ 2Þ 48 ) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 2 2 pffiffi jc1 j þ 2jc1 c2 j : 24K 2 ðtÞ tð1 þ tÞ a2 ¼
Let ð3:20Þ
ðn þ 2Þ 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 mb 1 þ Q1 þ pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ
! ¼ g1
ðsayÞ
Observe that if m b g1 , then the expression inside the first modulus of right-hand side (3.19) is non negative. Thus by applying Lemma 2 we get ð3:21Þ
jma22 a3 j a
ðn þ 1Þðn þ 2ÞQ1 12
3Q1 ðn þ 1Þm Q1 ðn þ 2Þ ! 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 pffiffi 24K 2 ðtÞ tð1 þ tÞ
which is the first part of the assertion (3.17).
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akshaya kumar mishra and priyabrat gochhayat Rewriting (3.18) jma22
ð3:22Þ
ð3:23Þ
( ðn þ 1Þðn þ 2ÞQ1 a3 j ¼ 2c2 Q1 1 48
) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm 2 þ c1 pffiffi ðn þ 2Þ 24K 2 ðtÞ tð1 þ tÞ ( ðn þ 1Þðn þ 2ÞQ1 ¼ 2c2 þ Q1 1 48 ) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm 2 þ c1 pffiffi ðn þ 2Þ 24K 2 ðtÞ tð1 þ tÞ ( ðn þ 1Þðn þ 2ÞQ1 a Q1 1 48 ) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm 2 þ pffiffi jc j þ 2jc2 j ðn þ 2Þ 1 24K 2 ðtÞ tð1 þ tÞ
Let, ð3:24Þ
ðn þ 2Þ 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 ma Q1 1 þ pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ
! ¼ g2
say
If m a g2 , then the expression inside the first modulus of right hand side of (3.23) is non negative. Thus by applying Lemma 2 we get ð3:25Þ
jma22 a3 j ¼
ðn þ 1Þðn þ 2ÞQ1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 Q1 þ pffiffi 12 24K 2 ðtÞ tð1 þ tÞ ! 3Q1 ðn þ 1Þm : ðn þ 2Þ
Which is the third part of assertion (3.17). Lastly (3.22) gives ðn þ 1Þðn þ 2ÞQ1 1 2 2 jma2 a3 j ¼ ð3:26Þ 2 c2 c1 2 48
! 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm 2 c1 þ Q1 þ pffiffi ðn þ 2Þ 24K 2 ðtÞ tð1 þ tÞ ( ðn þ 1Þðn þ 2ÞQ1 1 2 a 2 c 2 c 1 2 48 ) 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm 2 þ Q 1 þ pffiffi jc j : ðn þ 2Þ 1 24K 2 ðtÞ tð1 þ tÞ
fekete-szego¨ problem
323
Notably g2 < m < g1 gives 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 ðn þ 1Þm pffiffi a 1: Q 1 þ ðn þ 2Þ 24K 2 ðtÞ tð1 þ tÞ Therefore, if g2 < m < g1 , an application of Lemma 2 in (3.26) yields jma22
ðn þ 1Þðn þ 2ÞQ1 1 2 2 2c2 c1 þ jc1 j a3 j a 2 48 a
ðn þ 1Þðn þ 2ÞQ1 : 12
This is the middle part of the assertions in (3.17). The sharpness of each estimate in (3.17) can be established as in Theorem 1. The proof of Theorem 3 is complete. 9 Taking n ¼ 0 and n ¼ 1 in Theorem 3, we get the Fekete-Szego¨ inequalities for the classes k UCV and k SP respectively obtained recently by the present authors [12]. We list them here for the sake of completeness. Corollary 4. Let the function f given by (1.1) be in the class k UCV ðk > 1Þ. Then 8 ! 2 2 2 > Q 3Q m 4K ðtÞðt þ 6t þ 1Þ p > 1 1 > > Q1 ; m b g1 ; pffiffi > > 2 6 24K 2 ðtÞ tð1 þ tÞ > > > < Q1 2 jma2 a3 j a ; g2 a m a g1 ; > 6 > ! > > > > Q1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 3Q1 m > > ; m a g2 ; pffiffi > : 6 Q1 þ 2 24K 2 ðtÞ tð1 þ tÞ where KðtÞ is the complete elliptic integral of the first kind, Q1 is given in (2.1), ! 2 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 g1 ¼ 1 þ Q1 þ pffiffi 3Q1 24K 2 ðtÞ tð1 þ tÞ and 2 g2 ¼ 3Q1
! 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 Q1 1 þ : pffiffi 24K 2 ðtÞ tð1 þ tÞ
Each of the estimates is sharp.
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akshaya kumar mishra and priyabrat gochhayat
Corollary 5. Let the function f given by (1.1) be in ðk > 1Þ. Then 8 ! 2 2 2 > Q 4K ðtÞðt þ 6t þ 1Þ p > 1 > > 2Q1 m Q1 ; pffiffi > > 2 24K 2 ðtÞ tð1 þ tÞ > > > < Q1 jma22 a3 j a ; > 2 > ! > > > > Q1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 > > 2Q1 m ; pffiffi > : 2 Q1 þ 24K 2 ðtÞ tð1 þ tÞ
the class k SP m b g 1 ; g 2 a m a g1 ;
m a g 2 ;
where KðtÞ is the complete elliptic integral of the first kind, Q1 is given in (2.1), ! 1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 g1 ¼ 1 þ Q1 þ pffiffi 2Q1 24K 2 ðtÞ tð1 þ tÞ and g 2
! 1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 : ¼ Q1 1 þ pffiffi 2Q1 24K 2 ðtÞ tð1 þ tÞ
Each of the estimates is sharp. 4.
Remarks on main results
In this section we discuss some improvements of the middle inequalities in (3.5), (3.16) and (3.17) respectively. Remark 1. ð4:1Þ
The second part of assertion in (3.5) can be improved as follows: ðn þ 2Þ 1 ð1 A 2 Þð1 k 2 Þ 2 jma2 a3 j þ m ja22 j ðn þ 1Þ 3 18A 2 a
ðn þ 1Þðn þ 2ÞA 2 ; 6ð1 k 2 Þ
a2 a m a a3 ;
and ð4:2Þ
jma22
ðn þ 2Þ 1 ð5 þ A 2 Þð1 k 2 Þ þ a3 j þ m ja22 j ðn þ 1Þ 3 18A 2 a
ðn þ 1Þðn þ 2ÞA 2 ; 6ð1 k 2 Þ
where a3 is given by ð4:3Þ
a3 :¼
a3 a m a a1 ;
n þ 2 1 ð2 þ A 2 Þð1 k 2 Þ þ : nþ1 3 18A 2
fekete-szego¨ problem Proof. write
325
Suppose a2 a m a a3 . We continue with the estimate in (3.13) and
jma22 a3 j þ ðm a2 Þja2 j 2 ðn þ 2Þ 1 ð1 A 2 Þð1 k 2 Þ 2 ¼ jma2 a3 j þ m ja22 j ðn þ 1Þ 3 18A 2 ðn þ 1Þðn þ 2ÞA 2 1 2 2A 2 2 þ A2 6ðn þ 1ÞA 2 m 2 2 c 2 c 1 þ þ jc j a 2 ðn þ 2Þð1 k 2 Þ 1 24ð1 k 2 Þ ð1 k 2 Þ 3 ðn þ 2Þ 1 ð1 A 2 Þð1 k 2 Þ ðn þ 1Þ 2 A 4 c12 þ m 4ð1 k 2 Þ 2 ðn þ 1Þ 3 18A 2 ( ðn þ 1Þðn þ 2ÞA 2 1 2 2A 2 2 þ A2 6ðn þ 1ÞA 2 m a þ 2 c 2 c 1 þ jc 2 j 2 ðn þ 2Þð1 k 2 Þ 1 24ð1 k 2 Þ ð1 k 2 Þ 3 ) 6ðn þ 1ÞA 2 m 6A 2 1 ð1 A 2 Þð1 k 2 Þ þ jc12 j ðn þ 2Þð1 k 2 Þ ð1 k 2 Þ 3 18A 2 ðn þ 1Þðn þ 2ÞA 2 1 2 2 þ jc c a 2 c j 1 2 2 1 24ð1 k 2 Þ ¼
ðn þ 1Þðn þ 2ÞA 2 : 6ð1 k 2 Þ
We get (4.1). On the other hand suppose a3 a m a a1 . In this case we estimate jma22 a3 j þ ða1 mÞja2 j 2 . Now following the lines of proof for (4.1) with obvious changes we get (4.2). The proof of next two remarks are same as in Remark 1. Therefore we omit details. Remark 2. The second part of assertion in (3.16) can be improved as follows: n þ 2 1 p2 2ðn þ 1Þðn þ 2Þ 2 ð4:4Þ jma2 a3 j þ m ; b2 a m a b3 ; ja22 j a n þ 1 3 72 3p 2 and ð4:5Þ
jma22 a3 j þ
n þ 2 5p 2 1 2ðn þ 1Þðn þ 2Þ þ ; m ja22 j a n þ 1 72 3 3p 2
where b 3 is given by ð4:6Þ
b3 :¼
n þ 2 1 p2 þ : n þ 1 3 36
b3 a m a b1 ;
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akshaya kumar mishra and priyabrat gochhayat
Remark 3. The second part of assertion in (3.17) can be improved as follows: ( !) 2 2 2 ðn þ 2Þ 4K ðtÞðt þ 6t þ 1Þ p Q1 1 þ ð4:7Þ jma22 a3 j þ m ja22 j pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ a
ðn þ 1Þðn þ 2ÞQ1 ; 12
g2 a m a g3 ;
and (
ð4:8Þ
jma22
! ) ðn þ 2Þ 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 Q1 þ 1 þ a3 j þ m ja22 j pffiffi 3Q1 ðn þ 1Þ 24K 2 ðtÞ tð1 þ tÞ
ðn þ 1Þðn þ 2ÞQ1 ; 12 where g3 is given by a
ð4:9Þ
5.
g3 a m a g1 ;
! n þ 2 1 4K 2 ðtÞðt 2 þ 6t þ 1Þ p 2 þ g3 :¼ : pffiffi nþ1 3 72Q1 K 2 ðtÞ tð1 þ tÞ
An alternate method
Let Pðqk Þ denote the subclass of A, consisting of functions hðzÞ 0 qk ðzÞ in U, where qk ðzÞ is as in Lemma 1 and 0 denotes subordination. We need the following result: Lemma 3 [7]. Let k A ½0; yÞ be fixed and the function h A Pðqk Þ be given by the series expansion ð5:1Þ then ð5:2Þ
hðzÞ ¼ 1 þ b1 z þ b2 z 2 þ ;
ðz A UÞ
8 2 > u a 0; : Q1 ðkÞ þ ðu 1ÞQ12 ðkÞ; u b 1;
where Q1 ðkÞ is as in Lemma 1. hðzÞ ¼ qk ðz 2 Þ.
If 0 < u < 1 then equality holds in (5.2) if
Now replacing qk ðwðzÞÞ by hðzÞ in the equation (3.1) we get nþ1 a2 ¼ b1 2 a3 ¼
ðn þ 1Þðn þ 2Þ 2 ðb1 þ b2 Þ 12
fekete-szego¨ problem
327
and ma22
ðn þ 1Þðn þ 2Þ a3 ¼ 12
3ðn þ 1Þm 2 1 b1 b2 ðn þ 2Þ
An application of Lemma 3 with, u¼3
nþ1 m1 nþ2
yields ð5:3Þ
jma22 a3 j ¼
ðn þ 1Þðn þ 2Þ ðn þ 2Þ 2 nþ2 Q1 ðkÞ if < ma 12 3ðn þ 1Þ 3 nþ1
Putting the values of Q1 ðkÞ from Lemma 1 for 0 a k < 1, k ¼ 1 and k > 1 the estimate (5.3) simplifies to the middle inequalities in (3.5), (3.16) and (3.17) respectively. However, following the lines of proof Theorem 3.1 in [7] it can be shown that 2 nþ2 a1 ðkÞ; b1 ðkÞ; g1 ðkÞ > 3 nþ1 and 1 nþ2 a2 ðkÞ; b2 ðkÞ; g2 ðkÞ < 3 nþ1 for every 0 a k < y where a1 ðkÞ; a2 ðkÞ; b1 ðkÞ; b2 ðkÞ and g1 ðkÞ; g2 ðkÞ are defined as in Theorems 3.1, 3.2 and 3.3 respectively. Thus the methods of proof adopted for Theorems 3.1, 3.2 and 3.3 yield larger interval for m than the interval obtained for the estimates of jma22 a3 j in (5.3). References [ 1 ] N. I. Ahiezer, Elements of theory of elliptic functions (in Russian), Moscow, 1970. [ 2 ] G. D. Anderson, M. K. Vamanamurthy and M. K. Vuorinen, Conformal invariants, inequalities and quasiconformal maps, Wiley-Interscience, 1997. [ 3 ] B. C. Carlson and D. B. Shaffer, Starlike and pre-starlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), 737–745. [ 4 ] P. L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften 259, Bd., Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [ 5 ] M. Fekete and G. Szego¨, Eine bemerkung u¨ber ungerade schlichte funktionen, J. London Math. Soc. 8 (1933), 85–89. [ 6 ] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92. [ 7 ] S. Kanas, Coe‰cient estimates in subclasses of the Carathe´ory class related to conical domains, Acta Math. Univ. Comenian. 74 (2005), 149–161. [ 8 ] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327–336.
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[ 9 ] S. Kanas and A. Wisniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45 (2000), 647–657. [10] W. Koepf, On the Fekete-Szego¨ problem for close to convex functions, Proc. Amer. Math. Soc. 101(1) (1987), 89–95. [11] W. Ma and D. Minda, Uniformly convex functions I and II, Ann. Polon. Math. 57(2) (1992), 165–175, Ann. Polon. Math. 58(3) (1993), 275–285. [12] A. K. Mishra and P. Gochhayat, Fekete-Szego¨ problem for k-uniformly convex functions and for a class defined by the Owa-Srivastava operator, J. Math. Anal. Appl. 347 (2008), 563–572. [13] K. I. Noor, On new class of integral operator, J. Natur. Geom. 16 (1999), 71–80. [14] K. I. Noor and M. A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999), 341–352. [15] K. I. Noor and M. A. Noor, On certain classes of analytic functions defined by Noor integral operator, J. Math. Anal. Appl. 281 (2003), 244–252. [16] F. RØnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196. [17] H. M. Srivastava, A. K. Mishra and M. K. Das, A nested class of analytic functions defined by fractional calculus, Commun. Appl. Anal. 2 (1998), 321–332. [18] H. M. Srivastava and A. K. Mishra, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl. 39(3/4) (2000), 57–69. [19] H. M. Srivastava and S. Owa (eds.), Current topics in analytic function theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992. Akshaya Kumar Mishra Department of Mathematics Berhampur University Bhanja Bihar Berhampur-760007, Orissa India E-mail:
[email protected] Priyabrat Gochhayat Department of Mathematics Sambalpur University Jyoti Vihar Sambalpur-768019, Orissa India E-mail:
[email protected] 
