J Anal DOI 10.1007/s41478-016-0012-7 ORIGINAL RESEARCH PAPER
Coefficient multipliers on mixed-norm spaces Hðp; q; uÞ S. Naik1 • K. Rajbangshi1
Received: 11 May 2016 / Accepted: 2 November 2016 Ó Forum D’Analystes, Chennai 2017
Abstract The main purpose of this paper is to find necessary and sufficient conditions of coefficient multipliers from Hðp; q; u1 Þ to Hðs; t; u2 Þ spaces for certain ranges of the parameters p, q, s, t. We also find the sufficient conditions for coefficient multipliers on Hðp; q; aÞ by making use of the Gaussian hypergeometric functions F(a, b; c; z). Our results generalizes some well known results of Jevtic´, Pavlovic´, Duren and Shields, Ohya, Shimizu and Watanabe. Keywords Hypergeometric functions Coefficient multipliers Mixed-norm spaces Mathematics Subject Classification 33C05 44A35 42A45 26A33
1 Introduction Let HðDÞ denote the class of all analytic functions in the unit disc D of the complex plane. As usual, for 0\p\1 and f 2 HðDÞ, the integral mean of f Mp ðr; f Þ ¼
1 2p
Z
2p ih
p
jf ðre Þj dh
1p ;
0\r\1:
0
When p ¼ 1; and for such values of r,
& S. Naik
[email protected] K. Rajbangshi
[email protected] 1
Department of Applied Sciences, Gauhati University, Guwahati 781014, India
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S. Naik, K. Rajbangshi
M1 ðr; f Þ ¼ max jf ðreih Þj: 0 h\2p
For 0\p 1 and 0\a; q\1, the space Hðp; q; aÞ consists of those functions f 2 HðDÞ such that kf kp;q;a ¼
Z
1
ð1 rÞ
qa1
0
Mpq ðr; f Þdr
1q \1:
If q ¼ 1 and a 0; Hðp; 1; aÞ ¼ Hap equals the space of all f 2 HðDÞ satisfying kf kp;1;a ¼ kf kp;a ¼ sup ð1 rÞa Mp ðr; f Þ\1: 0\r\1
Notice that Hðp; p; 1pÞ ¼ Ap is the Bergman space, Hðp; 1; 0Þ ¼ H p is the Hardy 1 1 Þ ¼ Aðp; q; q1 Þ; is the space, Hð1; 1; 1p 1Þ ¼ Bp ; is the Besov space and Hðp; q; q1 weighted Bergman space. A positive continuous function u on [0, 1) is called normal if there is d 2 ½0; 1Þ and s and t, 0\s\t such that
uðrÞ uðrÞ is decreasing on ½d; 1Þ and lim s ¼ 0; r!1 ð1 rÞ ð1 rÞs uðrÞ uðrÞ t is increasing on ½d; 1Þ and lim t ¼ 1: r!1 ð1 rÞ ð1 rÞ For 0\p; q\1 and u normal, the mixed norm space Hðp; q; uÞ consists of those functions f 2 HðDÞ such that kf kHðp;q;uÞ ¼
Z
1 0
Mqp ðr; f Þ
up ðrÞ dr 1r
1=p \1:
If p ¼ 1; then Hð1; q; uÞ is the space of all f 2 HðDÞ satisfying kf kHð1;q;uÞ ¼ sup Mq ðr; f Þ\1: 0\r\1
aþ1
In particular, H ðq; p; ð1 rÞa Þ ¼ Hðp; q; aÞ and Hðq; p; ð1 rÞ p Þ ¼ Aðp; q; aÞ. Let XPand Y be two spaces of analytic functions on the unit disc D: P1Let f 2 Xn with 1 n f ðzÞ ¼ 1 a z : Suppose that fk g is a sequence such that n n n¼0 n¼0 n¼0 kn an z 2 Y: Then we can define the operator Tk : X ! Y as ðTk f ÞðzÞ ¼
1 X
kn an z n :
n¼0
Then the sequence fkn g1 n¼0 is said to be a coefficient multiplier or simply multiplier from X into Y. The set of multipliers from X to Y is denoted by (X, Y). Since the time of Hardy and Littlewood, mixed norm and related spaces have been used to study function spaces on D and later to study multipliers between such
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Coefficient multipliers on mixed-norm
spaces. Special emphasis has been put on the case where the spaces X and Y correspond to the sequence space of Taylor coefficient of analytic functions such as Hardy spaces, Bergman spaces, mixed norm spaces of analytic functions and so forth. The theory of the spaces Hðp; q; aÞ was originated due to the work of Hardy and Littlewood (1932, 1941). Their work on Hðp; q; aÞ spaces was continued by Flett and Sledd (see Flett 1972; Sledd 1978) and later on by Pavlovic´ (1986, 1987). Multipliers on Hardy spaces were in fashion for a long time and much work was done on them and related spaces. However nowadays complete descriptions of multipliers between Hardy spaces ðH p ; H q Þ for certain values of p and q remain still open. The reader is referred to the surveys (see Campbell and Leach 1984; Osikiewicz 2004) for lots of results and references. Many results on multipliers of Hðp; q; aÞ; H p ; Ap ; Bp and Aðp; q; aÞ have been established in the last decades (see Blasco 1995; Duren 1969; Duren and Shields 1970; Duren et al. 1969; Jevtic´ and Pavlovic´ 1998; Lou 1997; Mateljevic´ and Pavlovic´ 1990; Ohya et al. 1996; Wojtaszczyk 1990; Xiukui 2004; Naik and Rajbangshi 2014 and thereby). For this purpose of study, the use of Gaussian hypergeometric function is an important tool. In this article, we try to extend and generalize some results of Jevtic´, Pavlovic´, Duren and Shields and Ohya, Shimizu and Watanabe by an application of the Gaussian Hypergeometric functions. The standard ordinary fractional derivatives are replaced by hypergeometric transformation Da;b;c : The Classical/Gaussian hypergeometric function is defined by the power series expansion 2 F1 ða; b; c; zÞ
:¼ Fða; b; c; zÞ :¼ F ¼
1 X ða; nÞðb; nÞ zn
ðc; nÞ
n¼0
n!
ðjzj\1Þ:
Here a, b, c are complex numbers such that c 6¼ m; m ¼ 0; 1; 2; 3; . . . and (a, n) is the Pochhammer’s symbol/shifted factorial defined by Appel’s symbol ða; nÞ :¼ aða þ 1Þ ða þ n 1Þ ¼
Cða þ nÞ ;n 2 N CðaÞ
and ða; 0Þ ¼ 1 for a 6¼ 0 and C is the gamma function given by Z 1 CðzÞ ¼ et tz1 dt ReðzÞ [ 0: 0
Obviously, F(a, b; c; z) is analytic on the unit disc D. Many properties of the hypergeometric series including the Gauss and Euler transformations are found in standard textbooks such as (Andrews et al. 1999; Temme 1996). For any two analytic functions f and g represented by their power series expansion, f ðzÞ ¼
1 X n¼0
an zn ; gðzÞ ¼
1 X
bn z n
n¼0
in jzj R, let f g denote the Hadamard product (or convolution) of f and g defined by
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S. Naik, K. Rajbangshi
ðf gÞðzÞ ¼
1 X
an bn zn
ðjzj\R2 Þ:
n¼0
Equivalently, ðf gÞðrzÞ ¼
1 2p
Z
2p
f reiðthÞ gðreih Þdh;
0\r\1:
0
P n Let f ðzÞ ¼ 1 n¼0 an z 2 HðDÞ: For a, b, c non-negative real numbers we define socalled hypergeometric transformation Da;b;c f of f by Da;b;c f ðzÞ ¼ f ðzÞ Fða; b; c; zÞ ¼
1 CðcÞ X Cða þ nÞCðb þ nÞ n an z : CðaÞCðbÞ n¼0 Cðc þ nÞCð1 þ nÞ
For a ¼ 1, b ¼ 1 þ d, c ¼ 1, fractional derivative f ½d of order d [ 0 is given by f ½d ðzÞ ¼
1 X Cðn þ 1 þ dÞ n¼0
Cðn þ 1Þ
an z n :
In terms of convolution, the above mentioned definition can be rewritten as f ½d ðzÞ ¼ Cð1 þ dÞ½f ðzÞ Fð1; 1 þ d; 1; zÞ: P n m Let f ðzÞ ¼ 1 n¼0 an z 2 HðDÞ: The multiplier transformation D f of f, where m is any real number, is defined by Dm f ðzÞ ¼
1 X
ðn þ 1Þm an zn :
n¼0
In Jevtic´ and Pavlovic´ (1998) describe the multipliers between spaces Hðp; q; aÞ and either Hðs; t; bÞ or ls and complete some cases that were left in the previous known results on the study of this class of multipliers. The case 1\p\2 was open for many years although some partial answers were given in Jevtic´ (2013) and Jevtic´ and Pavlovic´ (1998). In Jevtic´ (2013) extend some results of Jevtic´ and Pavlovic´ (1998) and Pavlovic´ (1986). This motivates us to calculate the multipliers between the spaces Hðp; q; aÞ and Hðs; t; bÞ: We give a different proof by introducing socalled hypergeometric transformation and calculate the multipliers for 0\p\1 and also provide partial answer for the case 1\p\2: Also we find multipliers between the spaces Hðp; q; u1 Þ and Hðs; t; u2 Þ for certain values of the parameters p, q, s, t. Now onwards C denotes a positive constant independent of f, depending only on the parameters p; q; s; t; a; b. It is not necessarily the same on any two occurrences.
2 Preliminary results We recall some known results to be used in sequel.
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Coefficient multipliers on mixed-norm
Lemma 1
(Duren 1970, p. 65) For each p [ 1 and z ¼ reih ; Z 1 2p 1 dh Cð1 rÞðp1Þ : 2p 0 j1 zjp
Lemma 2
(Flett 1972, p. 757) If f 2 HðDÞ; then for 0\p; q\1 and 1\a\1, Z 1 ð1 rÞaþq Mpq ðr; f 0 Þdr\1 0
implies Z 0
Lemma 3
1
ð1 rÞa Mpq ðr; f Þdr\1:
(Shields and Williams 1971, p. 291) Suppose c [ 1 þ a [ 0. Then Z 1 ð1 rÞa ðc1aÞ ; 0\q\1: c dr Cð1 qÞ ð1 rqÞ 0
Lemma 4 (Flett 1972, p. 755) Suppose 0\p 1 and 0\a; q\1. If f 2 Hðp; q; aÞ; then Mp ðr; f Þ Cð1 rÞa :
Lemma 5 (Pavlovic´ 1988, p. g 2 H s ; 0\p s 1. Then
404)
Let
1
Ms ðr; f gÞ ð1 rÞ1p kf kp kgks ;
f 2 Hp; 0 p 1
and
0\r\1:
Lemma 6 (Mateljevic´ and Pavlovic´ 1990, p. 73) Let g be analytic in jzj\1, let m be a positive integer, let aj ¼ 0 for 0 j m, and let 0\p\1: Then C 1 r m Mp ðr; gðmÞ Þ Mp ðr; Dm gÞ Cr m Mp ðr; gðmÞ Þ; 0\r\1; where C does not depend on g. Lemma 7
(Duren 1970, p. 84) Let f(z) be analytic in jzj\1 and p1 p2 . Then 1
Mp2 ðr; f Þ Cð1 rÞp2
p1
1
Mp1 ðr; f Þ:
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S. Naik, K. Rajbangshi
3 Coefficient multipliers on Hðp; q; uÞ In this section we find necessary and sufficient conditions of coefficient multipliers on Hðp; q; uÞ spaces for certain ranges of the parameters p, q, s, t. Theorem 1 (a) (b)
Suppose that
0\q t\1 and pq st ; or 0\q 1 t\1 and pq 1 s\1:
Let m be an integer such that m þ 1 [ 1q þ b; where b is from the definition of the normality of the function u1 : Then g 2 ðHðp; q; u1 Þ; Hðs; t; u2 ÞÞ if and only if 1 u ðrÞ Mt r; gðmÞ C 1 ð1 rÞðmþ1qÞ : u2 ðrÞ P m n Proof Suppose f ðzÞ ¼ 1 n¼0 ðn þ 1Þ z : Let 0\r\1 be fixed and fr ðzÞ ¼ f ðrzÞ: Let g 2 ðHðp; q; u1 Þ; Hðs; t; u2 ÞÞ; then by closed graph theorem kfr gks;t;u2 Ckfr kp;q;u1 :
ð1Þ
Let 0\r\q\1. We have Z
1
us2 ðrÞ s Mt ðq; fr gÞdq r ð1 qÞ Z 1 us2 ðrÞ Mts ðr 2 ; Dm gÞ ð1 qÞsb1 dq ð1 qÞsb r
kfr gks;t;u2
¼ Cus2 ðrÞMts ðr 2 ; Dm gÞ: Since f ðzÞ ¼ Pm ðzÞð1 zÞm1 ; where Pm ðzÞ is a polynomial, from Lemma 1 it follows that Mqp ðq; fr Þ
1 ¼ 2p
Z 0
2p
1 j1 rzjðmþ1Þq
!pq dh
p
Cð1 rqÞqfðmþ1Þq1g
ifðm þ 1Þq [ 1:
Hence Z
up1 ðqÞ p Mq ðq; fr Þdq þ 0 1q ¼ I1 þ I2 :
kfr kpp;q;u1 ¼
123
r
Z
1 r
up1 ðqÞ p M ðq; fr Þdq 1q q
ð2Þ
Coefficient multipliers on mixed-norm
For m þ 1 [ 1q þ b, using Lemma 3 we get I1
Z
up1 ðrÞ ð1 rÞ
bp
r
p
ð1 qÞbp1 ð1 rqÞqfðmþ1Þq1g dq
0
Cup1 ðrÞð1
rÞ
ðmþ1Þpþpq
ð3Þ
:
For I2 I2
up1 ðrÞ pqfðmþ1Þq1g ap ð1 rÞ ð1 rÞ
Cup1 ðrÞð1
rÞ
ðmþ1Þpþpq
Z
1
ð1 qÞap1 dq
r
ð4Þ
:
The inequalities (1), (2), (3) and (4) together with Lemma 6 complete the proof of the first part. Conversely, let Mt ðr; gðmÞ Þ C
1 u1 ðrÞ ð1 rÞðmþ1qÞ ; u2 ðrÞ
ð5Þ
f 2 Hðp; q; u1 Þ and hðzÞ ¼ ðf gÞðzÞ: We need to prove that h 2 Hðs; t; u2 Þ: Using Lemma 2 it is sufficient to show that Z 1 ð6Þ us2 ðrÞð1 rÞsbþms1 Mts r; hðmÞ dr\1: 0
The monotone property of Mq ðr; f Þ gives that, if f 2 Hðp; q; u1 Þ; then Mq ðr; f Þ C Case I. Let 0\q t\1 and (7), we get
p q
1 : u1 ðrÞ
ð7Þ
st : By Lemmas 5, 6, inequality (5) and inequality
1 Mts ðr 2 ; hðmÞ Þ Cð1 rÞð1qÞs Mqs ðr; f ÞMts ðr; gðmÞ Þ
us1 ðrÞ ð1 rÞms Mqs ðr; f Þ us2 ðrÞ us ðrÞ ¼C 1s ð1 rÞms Mqsp ðr; f ÞMqp ðr; f Þ u2 ðrÞ up ðrÞ C 1s ð1 rÞms Mqp ðr; f Þ; u2 ðrÞ C
and hence last inequality gives (6). Case II. 0\q 1 t\1 and pq 1 s\1: By Lemmas 5, 6, 7, inequality (5) and inequality (7), we get
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S. Naik, K. Rajbangshi
Mts ðr 2 ; hðmÞ Þ CM1s ðr; f ÞMts ðr; gðmÞ Þ us ðrÞ C 1s ð1 rÞms Mqs ðr; f Þ u2 ðrÞ us ðrÞ ¼ C 1s ð1 rÞms Mqs1 ðr; f ÞMq1 ðr; f Þ u2 ðrÞ C
u11 ðrÞ ð1 rÞms Mq1 ðr; f Þ us2 ðrÞ
u11 ðrÞ ð1 rÞms Mq1q ðr; f ÞMpq ðr; f Þ us2 ðrÞ uq ðrÞ C 1s ð1 rÞms Mqq ðr; f Þ u2 ðrÞ uq ðrÞ ¼ C 1s ð1 rÞms Mqqp ðr; f ÞMqp ðr; f Þ u2 ðrÞ up ðrÞ C 1s ð1 rÞms Mqp ðr; f Þ: u2 ðrÞ ¼C
The above inequality gives (6). For p ¼ q; q ¼ p and u1 ðrÞ ¼ ð1 rÞ we get Corollary 2 (a) (b)
1þa p
1þb
and s ¼ t; t ¼ s and u2 ðrÞ ¼ ð1 rÞ s ;
(Ohya et al. 1996) Suppose that
0\p s\1 and qp st ; or 0\p 1 s\1 and qp 1 t\1:
Let 1\a; b\1 and m be an integer such that m þ 1 [ 1p þ 1þa q . Then g 2 ðAðp; q; aÞ; Aðs; t; bÞÞ if and only if 1 1þa 1þb Ms r; gðmÞ Cð1 rÞpþ q t m1 :
For u1 ðrÞ ¼ ð1 rÞa and u2 ðrÞ ¼ ð1 rÞb ; we have Corollary 3 (a) (b)
Suppose that
0\p s\1 and qp st ; or 0\p 1 s\1 and qp 1 t\1:
Let 1\a; b\1 and m be an integer such that m þ 1 [ 1p þ a: Then g 2 ðHðp; q; aÞ; Hðs; t; bÞÞ if and only if
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Coefficient multipliers on mixed-norm
1 Ms r; gðmÞ Cð1 rÞpþabm1 :
Corollary 4 (Duren and Shields 1970) Suppose that 0\p; s\1: Let m be the P n positive integer such that ðm þ 1Þ1 p m1 and let gðzÞ ¼ 1 n¼0 an z : Then g 2 p s ðB ; B Þ if and only if 1 1 M1 r; gðmÞ Cð1 rÞp sm :
4 Coefficient multipliers on Hðp; q; aÞ In this section using hypergeometric transformation we characterized the spaces of multipliers. In Theorem 5, we find the multipliers from Hðp; q; aÞ to Hðs; t; bÞ for certain values of the parameters. In Theorems 8 and 9, we describe the spaces of multipliers ðHap ; Hbs Þ: Theorem 5 Let 1 p; s\1; 0\q; t; a; b\1: Suppose a, b, c are non-negative real numbers with a þ b [ c and a þ b c 1p [ a: Then the following are true 1
(a)
If g 2 ðHðp; q; aÞ; Hðs; t; bÞÞ; then Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞþab :
(b)
If Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞþab ; g 2 ðHðp; q; aÞ; Hðs; t; b þ 1ÞÞ:
1
then
Proof Case (a). Let 0\r\1 be fixed and Fr ða; b; c; zÞ ¼ Fða; b; c; rzÞ Fr : If g 2 ðHðp; q; aÞ; Hðs; t; bÞÞ; then by the closed graph theorem kg Fr ks;t;b CkFr kp;q;a :
ð8Þ
For 0\r\1; kg Fr ks;t;b
Z
1
ð1 qÞ
tb1
r
Ms ðr; D
a;b;c
Mst ðq; Fr
Z
1
ð1 qÞ
gÞ
1t gÞdq tb1
1t dq
ð9Þ
r b
¼ Cð1 rÞ Ms ðr; Da;b;c gÞ: Since Fðc a; c b; c; zÞ is bounded on D for a þ b [ c; using Gauss identity (Andrews et al. 1999) and Lemma 1, we find
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S. Naik, K. Rajbangshi
Mpq ðq; Fr Þ
Z 2p qp
1 ih p
¼ F a; b; c; rqe dh 2p 0 Z 2p qp
1 ih ðcabÞp
ih p
¼ 1 rqe F c a; c b; c; rqe dh 2p 0 1 Cð1 rqÞððaþbcÞpÞq :
If a\ða þ b cÞ 1p, then Lemma 3 gives kFr kp;q;a C
Z
1
1q 1 ð1 qÞqa1 ð1 rqÞððaþbcÞpÞq dq
0
C ð1 rÞ½ððaþbcÞpÞq1qaþ1 1
1q
ð10Þ
1
¼Cð1 rÞpðaþbcÞþa : Inequalities (8), (9) and (10) give the desired result. Case (b). Suppose 1
Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞþab and f 2 Hðp; q; aÞ: From Lemma 5, we have Ms ðr 3 ; fr Da;b;c gr Þ CM1 ðr; f ÞMs ðr; Da;b;c gÞ: Using Lemma 4, we obtain 1
Ms ðr 3 ; fr Da;b;c gr Þ Cð1 rÞaþpðaþbcÞþab1 : For a ¼ b ¼ c ¼ 1; we have t
Mst ðr 3 ; fr gr Þ Cð1 rÞptbt : which implies f g 2 Hðs; t; b þ 1Þ: For q ¼ p; a ¼ 1p and t ¼ s; b ¼ 1s 1, we have Corollary 6 2
1
(a)
If g 2 ðAp ; As Þ; then Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞ s :
(b)
If Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞ s1 ; then g 2 ðAp ; As Þ:
2
1
1 1 and b ¼ t1 , we get For a ¼ q1
Corollary 7 Let 1 p; s\1; 0\q; t; a; b\1: Suppose a, b, c are non-negative real numbers with a þ b [ c and a þ b c 1p [ a: (i)
1
If g 2 ðAðp; q; aÞ; Aðs; t; bÞÞ; then Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞþab :
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Coefficient multipliers on mixed-norm
(ii)
1
If Ms ðr; Da;b;c gÞ Cð1 rÞpðaþbcÞþab ; beta þ 1ÞÞ:
then
g 2 ðAðp; q; aÞ; Aðs; t;
Next two results describe the coefficient multipliers spaces ðHap ; Hbs Þ for certain values of the parameters a and b: Theorem 8 Let 1 p; s\1; 0 a; b\1. For a, b, c non-negative real numbers with a þ b [ c we have (a)
If g 2 ðHap ; Hbs Þ; then Ms ðr; Da;b;c gÞ Cð1 rÞb :
(b)
s If Ms ðr; Da;b;c gÞ Cð1 rÞb ; then g 2 ðHap ; Haþb Þ:
Proof Case (a). Let 0\r\1 be fixed. If g 2 ðHap ; Hbs Þ; then by the closed graph theorem kg Fr ks;b CkFr kp;a C: Now kg Fr ks;b ¼ sup ð1 qÞb Ms ðq; g Fr Þ 0\q\1
Cð1 rÞb Ms ðr; Da;b;c gÞ: Therefore Ms ðr; Da;b;c gÞ Cð1 rÞb : Case (b). Suppose, Ms ðr; Da;b;c gÞ Cð1 rÞb and f 2 Hap : From Lemma 5, we have Ms ðr 3 ; fr Da;b;c gr Þ CM1 ðr; f ÞMs ðr; Da;b;c gÞ: Lemma 4 gives Ms ðr 3 ; fr Da;b;c gr Þ Cð1 rÞab : For a ¼ b ¼ c ¼ 1; we have Ms ðr; f gÞ Cð1 rÞab which completes the proof. Theorem 9 Let 0\p 1; 0\s; a; b\1. Suppose a, b, c non-negative real numbers with a þ b [ c; ða þ b cÞp [ 1 and a ¼ a þ b c: If g 2 ðHap ; Hbs Þ; 1
then Ms ðr; Da;b;c gÞ Cð1 rÞpab : Proof
Let 0\r\1 be fixed. If g 2 ðHap ; Hbs Þ; then by the closed graph theorem
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S. Naik, K. Rajbangshi
kg Fr ks;b CkFr kp;a :
ð11Þ
Since Fðc a; c b; c; zÞ is bounded on D for a þ b [ c; using Gauss identity (Andrews et al. 1999) and Lemma 1, we find
1 Mp ðq; Fr Þ ¼ 2p
Z
2p
ih ðcabÞp
j1 rqe j
ih
1 C 2p
Z 0
2p
1p
jFðc a; c b; c; rqe Þj dh
0
"
p
1 j1 rqeih jðaþbcÞp
#1p dh
1
Cð1 rqÞðapÞ : Therefore 1
kFr kp;a C sup ð1 qÞa ð1 rqÞpa 0\q\1
ð12Þ
1
Cð1 rÞpa : Again ð1 rÞb Ms ðr; Da;b;c gÞ kg Fr ks;b :
ð13Þ
Inequalities (11), (12) and (13) give the desired result. If a ¼ 1; b ¼ 1 þ d and c ¼ 1, then Da;b;c gðzÞ ¼ g½d ðzÞ. Hence from Theorem 9, we have Corollary 10
1
p ; Hbs Þ; then Ms ðr; g½d Þ Cð1 rÞpdb1 : If g 2 ðH1þd
Acknowledgements The first author wishes to acknowledge the financial support as principal investigator of the UGC project (University Grants Commission, India) (41-1387/2012 (SR)). Conflicts of interest We confirm that we have no potential conflict of interest.
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