The Modified Lommel functions: monotonic pattern and inequalities

THE MODIFIED LOMMEL FUNCTIONS: MONOTONIC PATTERN AND INEQUALITIES SAIFUL R. MONDAL

Abstract. This article studies the monotonicity, log-convexity of the modified Lommel functions by using its power series and infinite product representation. Same properties for the ratio of the modified Lommel functions with the Lommel function, sinh and cosh are also discussed. As consequence some Tur´ an type and reverse Tur´ an type inequalities are given. A Rayleigh type function for the Lommel

arXiv:1704.04667v1 [math.CA] 15 Apr 2017

functions is derived and as an application we obtain the Redheffer-type inequality.

1. Introduction The Lommel functions [12, 13] are the particular solution of the inhomogeneous Bessel differential equations x2 f′′µ,ν (x) + xf′µ,ν (x) − (ν 2 − x2 )fµ,ν (x) = xµ+1 ,

(1.1)

which are usually denoted as sµ,ν and Sµ,ν [4, 16] and Sµ,ν [17] given by    µ+ν+3 xµ+1 x2  Sµ,ν (x) = (µ+1) 1; µ−ν+3 , ; − 2 −ν 2 1 F2  2 2 4  µ−1 µ+ν+1 fµ,ν (x) := (1.2) Sµ,ν (x) = Sµ,ν (x) + (2i)iν Γ( µ−ν+1 )Γ(  2  2 )Jν (x)    S (x) = S (x) + i2µ−1 cos π(µ−ν) Γ( µ−ν+1 )Γ( µ+ν+1 )H (1) (x), µ,ν

µ,ν

2

2

2

ν

(1)

where Jν and Hν are respectively the Bessel and the Hankel function of the first kind. The above functions satisfy the recurrence relation
(1.3) fµ+2,ν (x) = xµ+1 − (µ + 1)2 − ν 2 fµ,ν (x)

The application of the Lommel functions can be seen in various branches of mathematics and mathematical physics. The mathematical properties of the Lommel functions are available in the literature [7, 11, 14, 17–21]. Like the modfied Bessel functions, the analogous of the Lommel functions is the modified Lommel functions. This functions first appear in the theory of screw propeller [15] and later analysed in [10, 17, 22]. The modified Lommel function gµ,ν is a particular solution of the differential equation (1.4)

x2 y ′′ (x) + xy ′ (x) − (x2 + ν 2 )y(x) = xµ+1 ,

and satisfies the relation gµ,ν (x) = i−(µ+1) fµ,ν (ix). Clearly, gµ,ν satisfies the recurrence relation
gµ+2,ν (x) = (µ + 1)2 − ν 2 gµ,ν (x) − xµ+1 . (1.5) For k ∈ {0, 1, 2, . . .}, consider the function

(1.6)

  µ−k+3 x2 ϕk (x) := 1 F2 1; µ−k+2 , , ; − 2 2 4

where x ∈ R and µ ∈ R such that µ − k is not in {0, −1, −2, . . .}. In [5], it is shown that ϕk is an even real entire function of order one and poses the Hadamard factorization ! ∞ Y x2 (1.7) , 1− 2 ϕk (x) = ηµ,k,n j=1 2010 Mathematics Subject Classification. 33C10; 26D7; 26D15. Key words and phrases. Lommel functions; Modified Lommel functions; Tur´ an-type inequality ; monotonicity properties; log-convexity .

1

2

SAIFUL R. MONDAL

where ±ηµ,k,n are all zeroes of ϕk . The infinite product in (1.7) is absolutely convergent. The function ϕk have close association with the Lommel function Sµ,ν as (1.8)

Sµ−k−1/2,1/2 (x) =

xµ−k+1/2 ϕk (x). (µ − k)(µ − k + 1)

For µ ∈ (0, 1), it is shown in [7] that Sµ−1/2,1/2 has only one zero in each of the interval     µ  µ π, (2n − 1 + µ) π , and I2n (µ) = 2nπ, 2n + π . I2n−1 (µ) = 2n − 1 + 2 2 In this article we consider the function Lµ,ν as   xµ+1 µ − ν + 3 µ + ν + 3 x2 Lµ,ν (x) := i−(µ+1) Sµ,ν (ix) = (1.9) . ; ; 1 F2 1; (µ − ν + 1)(µ + ν + 1) 2 2 4 The function Lµ,ν is known as the modified Lommel function. We also consider the normalized modified Lommel functions as ∞ X x2n
µ+ν+3
λµ,ν (x) = (µ − ν + 1)(µ + ν + 1)x−µ−1 Lµ,ν (x) = (1.10) . µ−ν+3 4n 2 2 n=0 n n

More details about the modified Lommel functions can be seen in [16, 22, 23] The Section 2 in this article is devoted for the investigation of the monotonicity properties of λµ,ν . Logconcavity and log-convexity properties in terms of the parameters µ and variable x are also investigated. As a consequence, direct and reverse Tur´an-type inequalities are obtained. The ratio of the derivatives of λµ,ν with sinh and cosh also considered in this section. In Section 3, the special case for the Lommel and the modified Lommel functions related to ϕk are considered. This section investigate the monotonicity and log-convexity for the product and the ratio of the Lommel and the modified Lommel functions. At the end a Redheffer-type inequality for both the Lommel and the modified Lommel functions is derived. Following lemma is required in sequel. P∞ P k k Lemma 1.1. [6] Suppose f (x) = ∞ k=0 ak x and g(x) = k=0 bk x , where ak ∈ R and bk > 0 for all k. Further suppose that both series converge on |x| < r. If the sequence {ak /bk }k≥0 is increasing (or decreasing), then the function x 7→ f (x)/g(x) is also increasing (or decreasing) on (0, r). Evidently, the above lemma also holds true when both f and g are even, or both are odd functions. 2. Monotonicity pattern

Theorem 2.1. Suppose that µ, µ1 > −1 and ν, ν1 ∈ R such that µ ± ν and µ1 ± ν1 are not negative odd integer. Then the following assertion are true. (i) Suppose that µ1 ≥ µ > −1 and (µ1 − µ)(µ1 + µ + 6) ≥ ν12 − ν 2 . Then, the function x 7→ λµ,ν (x)/λµ1 ,ν1 (x) is increase on (0, ∞). (ii) If µ ± ν + 3 > 0, then the function µ 7→ λµ,ν (x) is decreasing and log-convex on (−1, ∞) for each fixed ν ∈ R and x > 0. (iii) If µ ± ν + 3 > 0, then the function ν 7→ λµ,ν (x) is log-convex on R for each fixed µ > −1 and x > 0. (iv) The function x 7→ λ2k µ,ν (x)/ cosh(x) is strictly decreasing if (µ − ν + 3)(µ + ν + 3) > 2. (v) The function x 7→ λ2k+1 µ,ν (x)/ sinh(x) is strictly decreasing provided (µ − ν + 5)(µ + ν + 5) > 12. Proof. First consider a sequence {wn } defined by wn :=

(a − b)n (a + b)n , (c − d)n (c + d)n

where a, b, c, d are real numbers such that a ± b and c ± d are not negative integers or zero.

THE MODIFIED LOMMEL FUNCTIONS: MONOTONIC PATTERN AND INEQUALITIES

3

Then a calculation yield wn+1 (a − b + n)(a + b + n) = ≥ 1, wn (c − d + n)(c + d + n) provided a2 − b2 + 2an + n2 ≥ c2 − d2 + 2cn + n2 , which is equivalent to 2(a − c)n + a2 − b2 − c2 + d2 ≥ 0. The last inequality holds for all n ≥ 0 if a ≥ c and a2 − b2 − c2 + d2 ≥ 0. Choose a = (µ1 + 3)/2, b = ν1 /2, c = (µ + 3)/2 and d = ν/2. Then, a ≥ c is equivalent to µ1 ≥ µ and a2 − b2 − c2 + d2 ≥ 0 reduces to (µ1 − µ)(µ1 + µ + 6) > ν12 − ν 2 . This establish the fact that under the hypothesis in (i) the sequence {wn } is increasing. Since in this case {wn } represent the ratio of the coefficients of λµ,ν (x) and λµ1 ,ν1 (x), the result in (i) follows, in view of the Lemma 1.1. Two prove (ii) and (iii), consider the function

Γ µ−ν+3 Γ µ+ν+3 2 2

. gn (µ, ν) := + n Γ µ+ν+3 +n Γ µ−ν+3 2 2 The first and second partial differentiation of log(gn (µ, ν)) with respect to µ, ∂ log(gn (µ, ν)) = ∂µ ∂2 log(gn (µ, ν)) = ∂µ2






1 Ψ µ−ν+3 + n − Ψ µ+ν+3 +n , + Ψ µ+ν+3 − Ψ µ−ν+3 2 2 2 2 2




1 Ψ′ µ−ν+3 + n − Ψ′ µ+ν+3 +n + Ψ′ µ+ν+3 − Ψ′ µ−ν+3 2 2 2 2 4 ∂2 log(gn (µ, ν)). = ∂ν 2

Here, Ψ(x) = Γ′ (x)/Γ(x) is the digamma function which is increasing and concave on (0, ∞). Thus ∂ ∂µ gn (µ, ν)

gn (µ, ν)

=

∂ log(gn (µ, ν)) < 0 and ∂µ

∂2 ∂2 log(g (µ, ν)) = log(gn (µ, ν)) ≥ 0. n ∂µ2 ∂ν 2

This conclude that µ 7→ λµ,ν (x) is decreasing and log-convex on (−1, ∞). Also, ν 7→ λµ,ν (x) is log-convex on R for each fixed µ > −1 and x ∈ R. This prove (ii) and (iii) in view of the fact that the sum of log-convex functions is also log-convex. A computation yield λ(2k) µ,ν (x)

(2.1)

=

∞ X

µ−ν+3 2 n+k




n=0

It is well-known that

cosh(x) =

(2n + 2k)!
µ+ν+3

4n+k (2n)! n+k

2

x2n .

∞ X x2n . (2n)! n=0

In view of Lemma 1.1, it is enough to know the monotonicity of the sequence {αn }n≥0 where αn = Now, for all n ≥ 0 and k ≥ 0, the ratio αn+1 = αn 4 provided (µ − ν + 3)(µ + ν + 3) > 2. Similarly (2.2)

λ(2k+1) (x) = µ,ν

∞ X

n=0

2

n

4n+k

.

(2n + 2k + 2)(2n + 2k + 1)

< 1, + n + k µ+ν+3 +n+k 2

µ−ν+3 2

(2n + 2k + 2)!x2n+1
µ+ν+3 4n+k+1 (2n + 1)! 2 n+k+1 n+k+1


µ−ν+3 2

(2n + 2k)!
µ+ν+3


µ−ν+3 2 n

and

sin(x) =

∞ X x2n , (2n)! n=0

4

SAIFUL R. MONDAL (2k+1)

together with Lemma 1.1 yields that λµ,ν βn =

(x)/ sin(x) is decreasing if the sequence {βn }n≥0 where

µ−ν+3 2 n+k+1




(2n + 2k + 2)!
, µ+ν+3 4n+k+1 (2n + 1)! 2 n+k+1

is also decreasing. Again for all n ≥ 0 and k ≥ 0, the ratio βn+1 = βn 4

(2n + 2k + 4)(2n + 2k + 3)

< 1, + n + k + 1 µ+ν+3 +n+k+1 2

µ−ν+3 2

provided (µ − ν + 5)(µ + ν + 5) > 12. Hence the conclusion.



From Theorem 2.1, we have few interesting consequence. For example, the log-convexity of µ 7→ λµ,ν (x) means, for any α ∈ [0, 1] and for µ1 , µ2 > −1, 1−α λαµ1 +(1−α)µ2 ,ν (x) ≤ λα µ1 ,ν (x)λµ1 ,ν (x).

(2.3)

In particular, if µ1 = µ + a > −1 and µ2 = µ − a > −1 for µ, a ∈ R, and α = 1/2, then the above inequality gives the reverse of the Tur`an’s type inequality for the modified Lommel functions as λ2µ,ν (x) ≤ λµ+a,ν (x)λµ−a,ν (x). Similarly, the log-convexity of ν 7→ λµ,ν (x) gives λ2µ,ν (x) ≤ λµ,ν+a (x)λµ,ν−a (x). 3. Redheffer type bound In this section we prove the Redheffer-type inequality for some special kind Lommel and modified Lommel functions. From (1.7) and (1.8) it follows that ! ∞ x2 z µ+1/2 Y 1− 2 (3.1) , Sµ−1/2,1/2 (x) = (µ)(µ + 1) j=1 ηµ,0,n and this implies (3.2)

Lµ−1/2,1/2 (x) = i

−µ−1/2

∞ x2 z µ+1/2 Y 1+ 2 Sµ−1/2,1/2 (iz) = (µ)(µ + 1) j=1 ηµ,0,n

!

From (1.10) we have (3.3)

λµ−1/2,1/2 (x) = µ(µ + 1)z

−µ−1/2

Lµ−1/2,1/2 (x) =

∞ Y

1+

j=1

x2 2 ηµ,0,n

!

.

!

.

Also consider the normalized Lommel function as (3.4)

Λµ−1/2,1/2 (x) = µ(µ + 1)z

−µ−1/2

Sµ−1/2,1/2 (x) =

∞ Y

1−

j=1

x2 2 ηµ,0,n

Applying logarithmic differentiation on (3.3) gives λ′µ−1/2,1/2 (x)

(3.5)

zλµ−1/2,1/2 (x)

=

A calculation gives (3.6)

∞ X

2

η2 n=1 µ,0,n

+ x2

.

P∞ 2nz 2n−2 n=1 ( µ−ν+3 ) ( µ+ν+3 ) 4n λ′µ,ν (x) 2 2 2 n n . = lim P∞ = lim 2 − ν2 z 2n z→0 z→0 zλµ,ν (x) (µ + 3) n=0 ( µ−ν+3 ) ( µ+ν+3 ) 4n 2 2 n n

Now (3.5) and (3.6) together give the useful identity (3.7)

∞ X

1 4 1 = = . 2 2−1 η (2µ + 5) (µ + 2)(µ + 3) n=1 µ,0,n

.

THE MODIFIED LOMMEL FUNCTIONS: MONOTONIC PATTERN AND INEQUALITIES

5

For simplicity in sequel, we will use the notation ηµ,n for ηµ,0,n , the nth positive zero of ϕ0 (x). Next we state and proof some results involving the function λµ−1/2,1/2 and Λµ−1/2,1/2 . The monotonic properties of l’Hospital’ rule as state in the following result are useful in sequel. Lemma 3.1. [2, Lemma 2.2] Suppose that −∞ < a < b < ∞ and p, q : [a, b) 7→ ∞ are differentiable functions such that q ′ (x) 6= 0 for x ∈ (a, b). If p′ /q ′ is increasing(decreasing) on (a, b), then so is (p(x) − p(a))/(q(x) − q(a)) Next we will state and proof our main result in this section. Theorem 3.1. Suppose that µ > −1 and Iµ := (−ηµ,1 , ηµ,1 ). (1) The function x 7→ λµ−1/2,1/2 (x) is increasing on (0, ∞). (2) The function x 7→ λµ−1/2,1/2 (x) is strictly log-convex on Iµ and strictly geometrically convex on (0, ∞). (3) The modified Lommel functions λµ−1/2,1/2 (x) satisfies the sharp exponential Redheffer-type inequality ! aµ !bµ 2 2 ηµ,1 + x2 ηµ,1 + x2 (3.8) ≤ λµ−1/2,1/2 (x) ≤ , 2 − x2 2 − x2 ηµ,1 ηµ,1 2η 2

µ,1 are the best possible constant. on Iµ . Here, aµ = 0 and bµ = (µ+2)(µ+3) (4) The function x 7→ λµ−1/2,1/2 (x)Λµ−1/2,1/2 (x) is increasing on (−ηµ,1 , 0] and decreasing on [0, ηµ,1 ) (5) The function λµ−1/2,1/2 (x) Lµ−1/2,1/2 (x) x 7→ = , Λµ−1/2,1/2 (x) Sµ−1/2,1/2 (x) is strictly log-convex on Iµ . (6) The Lommel functions Λµ−1/2,1/2 (x) satisfies the sharp exponential Redheffer-type inequality !aµ !bµ 2 2 − x2 ηµ,1 ηµ,1 − x2 (3.9) ≤ Λµ−1/2,1/2 (x) ≤ , 2 2 ηµ,1 ηµ,1

on Iµ . Here, aµ = 0 and bµ =

2 2ηµ,1 (µ+2)(µ+3)

are the best possible constant.

Proof. Consider µ > −1 and x ∈ (−ηµ,1 , ηµ,1 ). (1) From (3.5) it is evident that (log(λµ−1/2,1/2 (x)))′ =

λ′µ−1/2,1/2 (x) λµ−1/2,1/2 (x)

=

∞ X

2x >0 + x2

η2 n=1 µ,n

on (0, ∞). Thus, for µ > −1, the function x 7→ log(λµ−1/2,1/2 (x)) is strictly increasing on (0, ∞) and consequently x 7→ λµ−1/2,1/2 (x) is also increasing on (0, ∞). (2) Again from (3.5) it follows that !′ ∞ 2 X λ′µ−1/2,1/2 (x) 2(ηµ,n − x2 ) = . 2 λµ−1/2,1/2 (x) (ηµ,n + x2 )2 n=1 Clearly, the function x 7→ λ′µ−1/2,1/2 (x)/λµ−1/2,1/2 (x) is increasing for x ∈ (−ηµ,1 , ηµ,1 ). This is equivalent to say that the function x 7→ λµ−1/2,1/2 (x) is log-convex on (−ηµ,1 , ηµ,1 ). Another calculation from (3.5) yields !′ ∞ 2 X zλ′µ−1/2,1/2 (x) 4zηµ,n = . 2 λµ−1/2,1/2 (x) (ηµ,n + x2 )2 n=1

This implies that the function x 7→ xλ′µ−1/2,1/2 (x)/λµ−1/2,1/2 (x) is strictly increasing for x ∈ (0, ∞) and hence x 7→ λµ−1/2,1/2 (x) is geometrically convex on (0, ∞).

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SAIFUL R. MONDAL

(3) Consider the function gµ (x) :=

log(λµ−1/2,1/2 (x)) 2 + x2 ) − log(η 2 − x2 ) . log(ηµ,1 µ,1

2 2 Denote p(x) := log(λµ−1/2,1/2 (x)) and q(x) := log(ηµ,1 + x2 ) − log(ηµ,1 − x2 ) on x ∈ [0, ∞). In view of (3.5), it follows that ∞ 4 4 − x4 ηµ,n − x4 λ′µ−1/2,1/2 (x) 1 X ηµ,1 p′ (x) . = = 2 2 q ′ (x) 4xηµ,1 λµ−1/2,1/2 (x) 2ηµ,1 η 2 + x2 n=1 µ,n

and then

d dx



p′ (x) q ′ (x)



=−

Thus, p′ (x)/q ′ (x) is decreasing. Therefore,

∞ 2 4 + ηµ,n x X z 4 + 2x2 ηµ,1 ≤ 0. 2 2 ηµ,1 (ηµ,n + x2 )2 n=1

gµ (x) =

p(x) p(x) − p(0) = q(x) − q(0) q(x)

is decreasing too on [0, ηµ,1 ) and hence aµ = lim gµ (x) < gµ (x) < lim gµ (x) = bµ . x→ηµ,1

x→0

Finally, lim

x→ηµ,1

p′ (x) =0 q ′ (x)

and

∞ 2 2 X ηµ,1 ηµ,1 1 p′ (x) = = , 2 x→0 q ′ (x) 2 n=1 ηµ,n 2(µ + 2)(µ + 3)

lim

2 implies aµ = 0 and bµ = ηµ,1 /(2(µ + 2)(µ + 3)).

(4) From (3.6) and (3.4), it is evident that  ∞  Y x4 1− 4 λµ−1/2,1/2 (x)Λµ−1/2,1/2 (x) = ηµ,n n=1

Thus, by the logarithmic differentiation it follows that  ′ λµ−1/2,1/2 (x)Λµ−1/2,1/2 (x)) ∞ X 4x3 . =− 4 λµ−1/2,1/2 (x)Λµ−1/2,1/2 (x) η − x4 n=1 µ,n

Since x ∈ Iµ , the conclusion follows. (5) From (3.4), we have the logarithmic differentiation of (Λµ−1/2,1/2 (x))−1 as log (Λµ−1/2,1/2 (x))−1 and log (Λµ−1/2,1/2 (x))−1



′′



′

=2

=

∞ X

2x 2 η − x2 n=1 µ,n

∞ 2 X ηµ,n + x2 > 0. 2 (ηµ,n − x2 )2 n=1

This conclude that the function x 7→ (Λµ−1/2,1/2 (x))−1 is strictly log-convex on Iµ . Finally, being the product of two strictly log-convex functions, the function x 7→

λµ−1/2,1/2 (x) Lµ−1/2,1/2 (x) = , Λµ−1/2,1/2 (x) Sµ−1/2,1/2 (x)

is also strictly log-convex. Note that the log-convexity of x 7→ λµ−1/2,1/2 (x) follows from part (2) of this theorem.

THE MODIFIED LOMMEL FUNCTIONS: MONOTONIC PATTERN AND INEQUALITIES

7

(6) To prove this result first we need to set up a Rayleigh type functions for the Lommel function. Define the function ∞ X (2m) −2m (3.10) αn,µ := ηµ,n , m = 1, 2, . . . . n=1

Logarithmic differentiation of (3.4) yield xΛ′µ−1/2,1/2 (x)

 −1 X ∞ ∞ ∞ X x2 x2 x2 X x2m x2 1 − = = −2 = . 2 Λµ−1/2,1/2 (x) η 2 − x2 η2 ηµ,n η2 η 2m n=1 µ,n n=1 µ,n n=1 µ,n m=0 µ,n ∞ X

Interchanging the order of the summation it follows that xΛ′µ−1/2,1/2 (x)

(3.11)

Λµ−1/2,1/2 (x)

= −2

Consider the function ϕµ (x) :=

(3.12)

∞ X ∞ ∞ X X x2m+2 (2m) 2m = −2 αn,µ x . 2m+2 η µ,n m=0 n=1 m=1

log(Λµ−1/2,1/2 (x)) p (x)   = µ . 2 qµ (x) log 1 − ηx2 µ,1

The binomial series together with (3.11) gives the ratio of p′µ and q′µ as p′µ (x) = q′µ (x)

(3.13)

xΛ′µ−1/2,1/2 (x) Λµ−1/2,1/2 (x) −2x2 2 ηµ,1

(2m)

2m Denote dm = ηµ,1 αn,µ . Then

dm+1 − dm =

 1−

2m+2 (2m+2) ηµ,1 αn,µ

x2 2 ηµ,1

−

P∞ (2m) 2m m=1 αn,µ x −1 = P∞ −2m 2m . m=1 ηµ,1 x

2m (2m) ηµ,1 αn,µ

∞ 2m X ηµ,1 = 2m η n=1 µ,n

2 ηµ,1 −1 2 ηµ,n

!

< 0.

This is equivalent to say that the sequence {dm } is decreasing, and hence by Lemma 1.1 it follows that the ratio p′µ /qµ′ is decreasing. In view of Lemma 3.1, we have ϕµ = pµ /qµ is decreasing. From (3.12) and (3.13), it can be shown that p′′µ (x) p′′µ (x) p′µ (x) 2 = lim ′′ = lim ′′ = ηµ,1 α(2) µ,n , ′ x→0 qµ (x) x→0 qµ (x) x→0 qµ (x)

lim ϕµ (x) = lim

x→0

and lim ϕµ (x) = lim

x→ηµ,1

x→ηµ,1

(2)

∞ 2 X p′µ (x) ηµ,1 − x2 = 1. = lim 2 − x2 qµ′ (x) x→ηµ,1 n=1 ηµ,n

2 It is easy to seen that ηµ,1 αµ,n = bµ .

 References

[1] M. Abramowitz and I. A. Stegun, A Handbook of Mathematical Functions, New York, (1965). [2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 (1993)no. 1. [3] G. E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge Univ. Press, Cambridge, 1999. [4] A. W. Babister, Transcendental functions satisfying nonhomogeneous linear differential equations, The Macmillan Co., New York(1967). [5] A. Baricz and S. Koumandos, Tur´an-Type Inequalities for Some Lommel Functions of the First Kind, Proc. Edinb. Math. Soc. (2) 59 (2016)no. 3. MR3572758 [6] M. Biernacki and J. Krzy˙z, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sklodowska. Sect. A. 9 (1955), 135–147 (1957).

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SAIFUL R. MONDAL

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