Third Hankel determinant for the inverse of reciprocal of bounded turning functions has a positive real part of order alpha Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 9. № 2 (2017). С. 112-121.

THIRD HANKEL DETERMINANT FOR THE INVERSE OF RECIPROCAL OF BOUNDED TURNING FUNCTIONS HAS A POSITIVE REAL PART OF ORDER ALPHA

B. VENKATESWARLU, N. RANI

Abstract. Let RT be the class of functions f (z) univalent in the unit disk E = z : \z\ < 1 such that Re f'(z) > 0 z e E, and H3( 1) be the third Hankel determinant for inverse f(z)

determinant, \t2t3 —14\, and the best possible upper bound for the third Hankel determinant H3(1) for the functions in the class of inverse of reciprocal of bounded turning functions having a positive real part of order alpha.

Keywords: univalent function, function whose reciprocal derivative has a positive real part, third Hankel determinant, positive real function, Toeplitz determinants.

Mathematics Subject Classification: 30C45; 30C50

1. Introduction Let A denote the class of all functions f (z) of the form

f (z) = z + ^ anzn (1.1)

n=2

in the open unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting of univalent functions. For a univalent function in the class A, it is well known that the nth coefficient is bounded by n. The bounds for the coefficients of univalent functions give information about their geometric properties. In particular, the growth and distortion properties of a normalized univalent function are determined by the bound of its second coefficient. The Hankel determinant of f for q ^ 1 and n ^ 1 was defined by Pommerenke [12] as

Hq (П)

an an+1 an+1 an+2

an+q-1 an+q

an+q-1 an+q

an+2q-2

This determinant has been considered by many authors. For example, Noor [10] determined the rate of growth of Hq (n) as n ^ ro for the bounded functions in S. Ehrenborg [4] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some of its properties were discussed by Layman in [7]. One can easily observe that the Fekete-Szego functional is H2(1). Fekete-Szego then further generalized the estimate for \a3 — ^a2\ with ^ real and f e S. R. M, Ali [1] found sharp bounds for the first four coefficients

B. venkateswarlu, N. RANI, third hankel determinant for the inverse of reciprocal of bounded turning functions has a positive real part of order alpha. © B. venkateswarlu, N. rani, 2017. Поступоила 29 июня 2016 г.

and sharp estimate for the Fekete-Szego functional I73 — ¿y2|, where t is real, for the inverse function of f defined as

f-1(w) = w + ^ YnWn,

n= 2

when it belongs to the class of strongly starlike functions of order a(0 < a ^ 1) denoted by ST (a). In the recent years several authors studied bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent analytic functions. In particular, for q = 2, n = 1, a1 = 1 and q = 2, n = 2, a1 = 1, the Hankel determinant simplifies respectively to

#2(1)

a1 a2 a,2 a3

= a3 — a2 and H2(2) =

a2 a3 a3 a4

a2a4 — a3.

For our discussion in this paper, we consider the Hankel determinant in the case of q = 3 and n = 1, denoted by H3(1), given by

a1 a2 a3

#3(1) = a2 a3 a4

a3 a4 a5

(1.2)

For f G A, ai = 1 we have

#3(1) = a3 (a2a4 — a3) — a4 (a4 — a2a3) + a5(a3 — a2)

and by applying the triangle inequality, we obtain

|#3(1)| ^ |a3||a2a4 — a3| + |a4||a2a3 — a4| + K||a3 — a21.

(1.3)

For the second Hankel functional H2(2) for the subclass RT of S consisting of functions whose derivative has a positive real part studied by Mac Gregor [9] the sharp upper bound was obtained by Janteng [6]. It was known that if f e RT then |ak| ^ f, for k E {2, 3, ■ ■ ■ }. Also the best possible sharp upper bound for the functional |a2a3 — a4| was obtained by Babalola [2] and hence the sharp inequality for |H3(1)|, for the class RT. Vamshee Krishna et al. [14] and also Venkateswarlu et al. [15] was obtained the sharp inequality |H3 (1)|, for the class of inverse of a function whose reciprocal derivative has a real part and of order alpha respectively. The sharp upper bound for the third Hankel determinant for the inverse of reciprocal of bounded turning functions was obtained by Venkateswarlu et al. [15].

Motivated by the above mentioned results obtained by different authors in this direction and the results by Babalola [2], in the present paper we seek an upper bound for the second Hankel determinant |t213 — t4| and hence an upper bound to the third Hankel determinant for certain subclass of analytic functions defined as follows.

Definition 1.1. A function f (z) e A is said to be function whose reciprocal derivative has a positive real part of order alpha (also called reciprocal of bounded turning function of order alpha) denoted by f e RT (a) for 0 ^ a ^ 1 if and only if

Re

1

f '(z )

> a, Vz e E.

(1.4)

Choosing a = 0, we obtain RT(0) = RT. Some preliminary lemmas required for proving our results are as follows.

2. Preliminary Results Let P denote the class of functions denoted by p such that

p(z) = 1 + ciz + C2Z2 + C3Z3 + ... = 1 + cnzn

(2.1)

n=1

which are regular in the open unit disc E and satisfy Re {p(z)} > 0 for each z g E. Here p(z) is called the Caratheodorv function [3].

Lemma 2.1. [11, 13] If p g P, then |ck| ^ 2, for each, k ^ 1 and the inequality i.s sharp for the function 1+Z.

Lemma 2.2. [5] The power series for p(z) given in (2.1) converges in the open unit disc E P

for n = 1, 2, 3,

2 Ci C2 ■ ■ Cn

C-1 2 Ci ■ ■ Cn-1

Dn = C-2 C-1 2 ■ ■ Cn-2

C—n C-n+1 C-n+2 ' ■ 2

are all non-negative. These determinants

p(z) = ^ Pk Po(eiifc z),

k=i

pk > 0, tk real and tk = tj, for k = j, where p0(z) = y+Z; in this case Dn > 0 for n < (m — 1) and Dn = 0 /or n ^ m.

This necessary and sufficient condition found in [5] is due to Caratheodorv and Toeplitz. We may assume without restriction that cY > 0. On using Lemma 2.2, for n = 2, we have

D2 =

which is equivalent to For n = 3,

2 Ci C2

cY 2 cY C2 Ci 2

[8 + 2Re {cic2} — 2 | C2 |2 — 4|ci |2] ^ 0,

2c2 = ci + x(4 — c2) for so me x, |x| ^ 1.

(2.2)

D3

2 C1 c2 c3

Ci 2 Ci C2

C2 Ci 2 Ci

C3 C2 Ci 2

> 0

22

(2.3)

and this is equivalent to

|(4C3 — 4CiC2 + C?)(4 — c2) + Ci(2C2 — c2)2| ^ 2(4 — C2)2 — 2|(2c2 — c1 (2.2) (2.3),

4c3 = c2 + 2c2(4 — c2)x — c2(4 — c2 )x2 + 2(4 — c2)(1 — |x|2)z

for some z wit h | z | ^ 1.

To obtain our results, we refer to the classical method initiated by Libera and Zlotkiewicz [8] and used then by several authors.

3. Main Result Theorem 3.1. Iff e RT (a)(0 < a < 1) and

f-1(w) = w + £ tnwn

n=2

in the vicinity of w = 0 is the inverse function of f, then

(1-a)2

\t2t4 —13\ 144

128a2-176a+137 a2-2a+2 2

3 (1 — a)

for 0 < a <

for 8 < a < 1,

3 8 ,

< a <

8

and the inequality is sharp. Proof For

ro

f (z) = z + Y anZn e RT(a),

n=2

there exists an analytic function p e P in the open unit disc E with p(0) = 1 and Rep(z) > 0 such that

1 _ a f(z)

--= p(z) ^ 1 — af' (z) = (1 — a)f '(z)p(z). (3.1)

(1 — a)z/(z)

Replacing f'(z^d p(z) with their equivalent series expressions in (3.1), we have

ro ro ro

1 — a (1 + Y nanzn-^ = (1 — a) (1 + Y na„zn) (1 + Y Cnzn

n=2 n=2 n=1

Upon simplification, we obtain

(1 — a) — 2aa2z — 3aa3z2 — 4aa4z3 — 5aa5z4 — ... = (1 — a) + z(1 — a)(c1 + 2a2)

+ z2(1 — a)(c2 + 2a2C1 + 3a3) + z3(1 — a)(c3 + 2a2C2 + 3a3C1 + 4a4) (3.2)

+ z4(1 — a)(c4 + 2a2C3 + 3a3C2 + 4a4C1 + 5a5)----

Equating the coefficients at like powers of z, z2, z3 and z4 respectively on both sides of (3.2), after simplifying, we get

c1(1 — a)

a2

2

(1 — a) 2

a3 =--3—[c2 — (1 — a)ciJ;

a4 a5

(1 — a)

4

(1—a

5

(3.3)

C3 — 2(1 — a)c1d + (1 — a)2c1 C4 — 2(1 — a)c1c3 + 3(1 — a)2c?c2 — (1 — a)c2 — (1 — a)3c1

Since

f (z) = z + Y anzn e RT(a),

n=2

f,

ro ro ro / ro \ n

W = f (f-1(w)) = f-1 (w) + Y an(f-1(w))n ^ w = w + Y tnwn + Y aj w + Y tnWn j

n=2 n=2 n=2 n=2

n=2 n=2 n=2 n=2

After simplifying, we get

(t2 + a2)w2 + (t3 + 2a2t2 + a3)w3 + (t4 + 2a2t3 + a2t2 + 3a3t2 + a4)w4 + (t5 + 2a2t4 + 2a2t2t3 + 3a313 + 3a3t2 + 4a4t2 + a5 )w5 +

0.

(3.4)

Equating the coefficients of like powers of w2,w3,w^d w5 on both sides of (3.4), respectively, further simplification gives

t2 — —a2; t3 — —a3 + 2a2; t4 — —a4 + 5a2a3 — 5a3; t5 = —a5 + 6a2a4 — 21a2a3 + 3a3 + 14a^

(3.5)

a2 a3 a4 a5

t2 t4 t5

C1 (1 — a) 2 ; (1 — a)

t3

(1 — a) 6

2C2 + (1 — a)c1

24 (1 — a) 120

6C3 + 8(1 — a)c1d + (1 — a)2c1

(1 — a)3c1 + 42C1C3(1 — a) + 16c2(1 — a) + 22^(1 — a)2 + 24c4

(3.6)

Substituting the values of t2, t3 and t4 from (3,6) in the funetional \t2t4 — t3\ for the function f e RT(a), upon simplification, we obtain

\t2t4 — t2\

which is equivalent to

2, (1 — a)2

144

18C1C3 + 8(1 — a)c1c2 — 16c2 — (1 — a)2c1

\t2t4 — t2\

(1 — a)2

144

d1C1C3 + d2C^C2 + d3C2 + ¿4c1

d1 = 18; d2 = 8(1 — a); d3 = —16; d4 = —(1 — a)2

(3.7)

(3.8)

C2 C3

\z\ < 1,

4\d1C1C3 + d2c1c2 + d3c2 + d4c1\ < (¿1 + 2d2 + ^3 + 4d4)c1 + 2^d(4 — c£)

+ 2(d1 + ¿2 + d3)c1(4 — c2)\x\ — {(¿1 + ¿3)^ + 2d1C1 — 4d3>\x\2(4 — c2) From (3.8) and (3.9), we can now write

d1 + 2d2 + d3 + 4d4 = 2(—2a2 — 4a + 7); 2(d1 + d2 + d3) = 4(5 — 4a);

(d1 + d3)c2 + 2d1C1 — 4d3 = 2(c1 + 18c1 + 32); d1 = 18.

(3.9)

(3.10)

Since c1 = c e [0, 2], using the result (c1 + a)(c1 + b) ^ (c1 — a)(c1 — b), where a,b ^ 0, and applying triangle inequality, we can have

— (d1 + d3)c1 + 2d1C1 — 4d3 = —2(c2 — 18c1 + 32).

(3.11)

Substituting the calculated values from (3.10) and (3.11) on the right-hand side of (3.9), we have

4\d1C1C3 + d2c1c2 + d3c2 + ¿4c4\ < 2(—2a2 — 4a + 7)c1

+ 36(4c1 — c3) + 4c1(5 — 4a)(4 — c2)\x\ — 2(c2 — 18c1 + 32)\x\2(4 — c1)

(3.12)

Choosing ci = c G [0, 2], applying the triangle inequality and replacing |x| by ^ on the right-hand side of the above inequality

4|diCiC3 + d2C?C2 + 4c2 + 4c4| ^ 2(—2a2 — 4a + 7)c4 + 36c(4 — C2)

+ 4c2(5 — 4a)(4 — c2)j + 2(c — 16)(c — 2)j2 (4 — c2) = F(c,j), 0 ^ j = |x| ^ 1 and 0 ^ c ^ 2.

(3.13)

Now we maximize the function F(c, on the closed region [0, 2] x [0,1]. Differentiating F(c, given in (3.13) with respect to we obtain

dF _ 4

(5 — 4a)c2 + (c — 16)(c — 2)j (4 — c2) > 0.

(3.14)

For 0 < ^ < 1 and for fixed c with 0 < c < 2, from (3.14), we observe that > 0. Therefore, F(c, becomes an increasing function of ^ and hence it cannot have a maximum value at any point in the interior of the closed region [0, 2] x [0,1]. Moreover, for a fixed c G [0, 2], we have

max F(c,^) = F(c 1) = G(c).

Therefore, replacing ^ by 1 in F(c, upon simplification, we obtain

G(c) = -4(a2 - 2a + 2)c4 + 8(3 - 8a)c2 + 256. G'(c) = —16(a2 - 2a + 2)c3 + 16(3 - 8a)c. G''(c) = -48(a2 - 2a + 2)c2 + 16(3 - 8a). For optimum value of G(c), consider G'(c) = 0. From (3.16), we get

2 r 3 - 8a c = 0 or c2

La2 — 2a + 2J

(3.15)

(3.16)

(3.17)

(3.18)

Case 1. Suppose 0 ^ a < §. By (3.18), for c = 0, in (3.17), we get G"(c) = 16(3 — 8a) > 0. Then G(c) has minimum at c = 0. For c2 = a23--§t"+2, in (3.17), we get G"(c) = —32(3 — 8a) < 0. Therefore, by the second derivative test, G(c) has maximum value at

3 - 8a a2 - 2a + 2. c

G(c)

4(128a2 - 176a + 137)

Gmax(c) a2 — 2a + 2 '

Simplifying expressions (3.13) and (3.19), we have

, 2 , 2 , 4, 128a2 — 176a + 137

|diCiC3 + d2CiC2 + d3C2 + ^cj ^ -2—-——-.

a2 — 2a + 2

By relations (3.7) and (3.20), upon simplification, we obtain

|t2t4 — t3| =

(1 — a)2 144

r128a2 — 176a + 1371 a2 — 2a + 2

(3.19)

(3.20)

(3.21)

Case 2. Suppose § ^ a ^ 1. By (3.18), for

3 8a

a2 2a + 2

C

in (3,17), we get G"(c) = -32(3 — 8a) > 0. Then G"(c) has minimum at

3 - 8a

a2 — 2a + 2'

For c = 0, in (3.17) we get G"(c) = 16(3 — 8a) < 0. Therefore, by the second derivative

test, G(c) has maximum value at c = 0. Substituting the value of c in expression (3,15), upon simplification, we obtain the maximum value of G(c):

Gmax(c) = 256. (3.22)

Simplifying the expressions (3.13) and (3.22), we obtain

|dicica + d2c2c2 + dac2 + ¿^41 ^ 64. (3.23)

From the relations (3.7) and (3.23), upon simplification, we obtain

|t2t4 — ¿31 =

3(1 — a)

By setting c1 = c = 0 and choosing x = 1 in expressions (2.2) and (2.4), we find that c2 = 2

c3 = 0

sharp and the extremal function in this case is given by

1

f '(z)J

1 + 2z2 + 2z4 +

1 + z2 1- z2

This completes the proof of our theorem.

□

Remark 3.1. For this we chose a = 0, in (3.21) we get |t2t3 — t|| ^ U|. This result is coincides with Venkateswarlu et al. [15] and also Vamshee Krishna et al. [14]. From this we conclude that, for a = 0, the sharp upper bound for the second Hankel determinant of a function whose derivative has a positive real part and a function whose reciprocal derivative has a positive real part is the same.

Theorem 3.2. Iff e RT (a)(0 ^ a ^ 1) and

<x

f-i(w) = w + J2

n=2

in the vicinity of w = 0 is the inverse fund ion of f, then

2

| ¿2*3 — t4 3

1 — a /13 — 4a ^ 2 Va2 — 2a + 4 V 6

Proof. Substituting the values of t2,t3 and t4, from in the determinant |t2t3 — t4| for the function f e RT(a), after simplifying, we get

| ¿2*3 — ¿41

(1 a)2 (2cic2 + (1 — a)c3) — a)(6c3 + 8(1 — a)c^ + (1 — a)2c3)

12 (1 — a) 24

24

|(1 — a)2c3 — 6c3 — 4cic2(1 — a)|.

(3.24)

c2 c3

| z| < 1 ,

2 | (1 — a)2c? — 6c3 — 4cic2(1 — a) — (5 — 2a2)c? — 6(4 — c?)

— 2ci(4 — ci)(5 — 2a) | x | +3(ci + 2)(4 — ci) | x |2

(3.25)

2

2

Since ci = c G [0, 2], using the estimate (ci + a) ^ (ci - a), where a ^ 0, applying the triangle inequality and replacing |x| by ^ in the right-hand side of the above inequality, we have

2 | (1 — a)2ci — 6c3 — 4c2c2(1 — a) (5 — 2a2)c3 + 6(4 — c2)

+ 2c(4 — c2)(5 — 2a)j + 3(c — 2)(4 — c2)j2 = F(c,j), 0 ^ j =| x 1 and 0 ^ c ^ 2.

(3.26)

Then we maximize the function F(c, on the closed square [0, 2] x [0,1]. Differentiating (3.26), i.e., F(c, with respect to we get

dF

(5 — 2a)c + 3(c — 2)j (4 — c2) > 0.

As described in Theorem 3.1, further, we obtain

G(c) = -2c3(a2 - 2a + 4) + 4c(13 - 4a). G'(c) = -6c2(a2 - 2a + 4) + 4(13 - 4a). G''(c) = - 12c(a2 - 2a + 4). For optimum value of G(c), consider G'(c) = 0. From (3.28), we get

(3.27)

(3.28)

(3.29)

2(13 — 4a) 3(a2 — 2a + 4)

2(13-4a)

for 0 ^ a ^ 1.

Using the obtained value of c = ^3(0i-2a+>4) G [0, 2] in (3.29), we arrive

at

G"(C) = —12

2(13 — 4a) 3(a2 — 2a + 4)

(a2 — 2a + 4) < 0.

G(c) c

2(13 — 4a) 3(a2 — 2a + 4) '

c

G(c) c

Gma,x

32

Va2 — 2a + 4

13 — 4a 6

By expressions (3.26) and (3.30), after simplifying, we get

| (1 — a)2ci — 6C3 — 4(1 — a)cic2

16

Va2 — 2a + 4

r13 — 4a1 6

Simplifying the relations (3.24) and (3.31), we obtain

11213 — t4| ^

2(1 — a)

Va2 — 2a + 4

13 — 4a 6

This completes the proof of the theorem.

(3.30)

(3.31)

(3.32)

□

This

Remark 3.2. For the choice of a = 0, from (3.32), we obtain |t2t3 - t4| ^ ] i

result coincide with that by Vamshee Krishna et al. [14] and also by Venkateswarlu et al. [16]. We observe that the upper bound for |t2t3 -14| of a function whose derivative has a positive real part [14] and a function whose reciprocal derivative has a positive real part is the same.

2

2

C

C

3

2

3

The next theorem can be proved straightforwardly by applying the same procedure as in the

ci = 0 c2 = 2 x = 1 .

ro

Theorem 3.3. If f e RT(a)(0 ^ a ^ 1) and f-i(w) = w + tnwn near w = 0 is the

n=2

inverse function of f then |t3 — i|| ^ 2[1 — a].

Using the fact that

|cn| ^ 2, n e N = {1, 2, 3,...},

c2 c3 arrive at the following inequalities.

Theorem 3.4. Iff (z) e RT (a), (0 ^ a ^ 1) and

oo

f-i(w) = w + £ tnwn

n=2

w = 0 f

(i) |¿21 ^ (1 — a),

(n) |t31 ^ I(1 — a)(2 — a),

(Hi) |¿41 ^ [2a2 — 12a + 13],

(iv) |¿51 ^ [—2a3 + 28a2 — 79a + 59]

hold.

Using the results of Theorems 3,1, 3,3, 3,5 and 3,6, we arrive at the following corollary,

ro

Corollary 1. If f e RT(a)(0 ^ a ^ 1) and f-i(w) = w + ¿nwn near w = 0 is the

f

n=2

|H3(1)|

(1 — a)2 1080

5(a2 — 3a + 2)(128a2 — 176a + 137) a2 — 2a + 2

120(2a2 — 12a + 13) /13 — 4a^ 2 + Va2 — 2a + 4 V 6

+ 48(—2a3 — 28a2 — 79a + 59)

(1 — a)2 135

40(a2 - 3a + 2) +

15

Va2 — 2a + 4 V 6

for 0 ^ a < -, j ^ 8'

(13 — 4a\2

+ 6(—2a3 — 28a2 — 79a + 59)

for - ^ a ^ 1. j 8

Remark 3.3. If we choose a = 0 in the above expressions, we obtain |H3(1)| ^ 0.742. These inequalities are sharp and coincide with the results by Vamshee Krishna et al. [14] and also by Venkateswarlu et al. [15]. We observe that the upper bound for the third Hankel determinant of a function whose derivative has a positive real part [14] and a function whose reciprocal derivative has a positive real part is the same.

The authors thank D, Vamshee Krishna for discussing particular aspects of the work.

СПИСОК ЛИТЕРАТУРЫ

1. R. M. Ali. Coefficients of the inverse of strongly starlike functions, // Bull. Malays. Math. Sci. Soc. 26:1, 63-71 (2003).

2. К. O. Babalola. On H3( 1) Hankel determinant for some classes of univalent functions //in "Inequality Theory and Applications" 6 , 1-7 (2010).

3. P. L. Duren. Univalent functions. Grundlehren der Mathematischen Wissenschaften. 259. Springer, New York (1983).

4. R. Ehrenborg. The Hankel determinant of exponential polynomials // Amer. Math. Monthly. 107:6, 557-560 (2000).

5. U. Grenander and G. Szego. Toeplitz forms and their applications. Chelsea Publishing Co., New York (1984).

6. A. Janteng, S.A. Halim and M. Darus. Coefficient inequality for a function whose derivative has a positive real part //J. Inequal. Pure Appl. Math. 7:2, 1-5 (2006).

7. J. W. Layman. The Hankel transform and some of its properties // J. Integer Seq. 4:1, 1-11 (2001).

8. R.J. Libera and E.J. Zlotkiewicz. Coefficient bounds for the inverse of a function with derivative in P // Proc. Amer. Math. Soc. 87:2, 251-257 (1983).

9. Т.Н. Mac Gregor. Functions whose derivative have a positive real part // Trans. Amer. Math. Soc. 104:3, 532-537 (1962).

10. K.I. Noor. Hankel determinant problem for the class of functions with bounded boundary rotation 11 Rev. Roum. Math. Pures Appl. 28:8, 731-739 (1983).

11. Ch. Pommerenke. Univalent functions. Vandenhoeck and Ruprecht, Gottingen (1975).

12. Ch. Pommerenke. On the coefficients and Hankel determinants of univalent functions //J- Lond. Math. Soc. sl-41:l, 111-122 (1966).

13. B. Simon, Orthogonal polynomials on the unit circle, part 1. Classical theory. Part. 1. AMS Colloquium Publ. 54. American Mathematicical Socetv, Providence, RI (2005).

14. D. Vamshee Krishna, B. Venkateswarlu and T. RamReddv. Third Hankel determinant for the inverse of a function whose derivative has a positive real part // Matematvchni Studii. 42:1, 54-60 (2014).

15. B. Venkateswarlu, D. Vamshee Krishna and N. Rani. Third Hankel determinant for the inverse of reciprocal of bounded turning functions,// Buletinul academiei de stiinte a Republicii Moldova. Matematica. 79:3, 50-59 (2015).

16. B. Venkateswarlu, D. Vamshee Krishna and N. Rani. Third Hankel determinant for the of reciprocal of bounded turning functions has a positive real part of order alpha // Studia Universitatis Babe-Bolvai Mathematica, to appear.

B. Venkateswarlu, Department of Mathematics, GST, GITAM University, Benguluru Rural Dist-562 163, Karnataka, India E-mail: bvlmaths@gmail.com

N. Rani,

Department of Sciences and Humanities,

Praveenva Institute of Marine Engineering and Maritime studies, Modavalasa- 534 002, Visakhapatnam, A. P., India E-mail: raninekkantillll@gmail. com