ZALCMAN CONJECTURE AND HANKEL DETERMINANT OF ORDER THREE FOR STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH SHELL-LIKE CURVES

Probl. Anal. Issues Anal. Vol. 9 (27), No 2, 2020, pp. 119-137

DOI: 10.15393/j3.art.2020.6950

UDC 517.546, 517.547

V. SüMAN Kumar, R. Bharayi Sharma

ZALCMAN CONJECTURE AND HANKEL DETERMINANT OF ORDER THREE FOR STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH SHELL-LIKE CURVES

Abstract. The aim of this article is to estimate an upper bound of |#3(1)|, the Zalcman coefficient functional for n = 3 and n = 4, and also to investigate the fifth, sixth, seventh coefficients of starlike and convex functions associated with shell-like curves. Similar type of outcomes are estimated for the functions f-1 and .

f(z) = z + ^anzn Wz eU

in the unit disc U = [z E C : | z |< 1}. Let S be a subclass of A consisting of univalent functions in U. A function f E S is a starlike function iff

ne{zj\\z)H(z)) > 0; Vz eU.

A function f E S is a convex function iff

Ke(l + (zf"(z)/f (z))) > 0; Vz eU.

© Petrozavodsk State University, 2020

Let V be the family of holomorphic functions p in U with the conditions {p (z)} > 0 and p (0) = 1 represented in the form

p (z) = 1 + C\\Z + C2Z2 + C3Z3 + ... (2)

For f E S, Lawrence Zalcman conjectured that —

I < (n — 1)2.

This conjecture was proved by Krushkal [10] for n = 3, 4, 5, 6. It was also considered by Ma [11] and Ravichandran [14]. Equality holds for the Koebe function and its rotations. A holomorphic function F is subordinate to another holomorphic function h, denoted by F — h, iff there exists a Schwarz function w in U with the conditions w (0) = 0, and Iw(z)I — 1 < 0, such that F (z) = h (w (z)). The Hadamard product of two holomorphic

functions f (z) = z + anzn and g(z) = z + gnzn in A, is defined as

(f*g)(z) = z + an9nZn.

In the year 1966, Pommerenke [12] has denoted the Hankel determinant by Hq (n) and defined it as

n+1 n+2 n+3 ...... n+q

an+q-1 an+q an+q+1 ...... an+2q-2

Here n, q E N, (an) is a sequence of real or complex numbers. For different values of n and q, one obtains different Hankel determinants and also some particular cases of Fekete-Szego coefficient functional. Many authors [1], [4], [6], [15], [16], [18] have studied the Fekete-Szego coefficient functional for different subclasses of univalent, multivalent, and holomorphic functions. For n = q = 2, the relation (3) reduces to IH2(2)I= |a2a4 — a2|. This is known as the second-order Hankel determinant. Several authors [7], [8], [18] have studied this determinant for different subclasses of holomorphic functions. For n = 1 and q = 3, from the determinant Hq (n) after applying the triangle inequality, one gets an upper bound for the third order Hankel determinant, given by

IH3 (1) I< MIa2«4 — a3I + MIa4 — «2^31 +1«5II«3 — ^I

This is known as the third Hankel determinant for a1 = 1. Babalola [3], Srivastava [17], Vamshee Krishna and Ram Reddy [18] have studied the third order Hankel determinant H3 (1) for different subfamilies of analytic functions.

Raina and Sokol [4], [13] have used the function q(z) as the superordi-nating function. The function q (z) = Vl + z2+z is analytic and univalent on C \\ [i, -i}, which maps the unit disc onto a shell shaped region on the right half plane. It is symmetric with respect to the real axis from 0.4 to 2.41. It is a function with positive real part with q (0) = q& (0) = 1.

Figure 2: Shell shaped region

Using q(z ), they have defined S * (q) as shown below and studied the initial coefficients, Fekete-Szego coefficient functional, Hankel determinant of order two for the function f in S * (q).

Definition 1. f G A is a function of the class S * (q) iff

zf& (z) r--4TV W1 + z2 + z. f (z)

Definition 2. f,g G A are two functions of the class S* (q) iff

* ( (f * g)&(z) ) ._

——VTT^2 + Z. (f * g) (z)

Definition 3. f G A is a function of the class C (q) iff

(l + ^VÏ+72 + z. (5)

^ J (z) &

Subordination results of this kind for various subclasses of analytic functions were obtained by several mathematicians, e.g., [2], [5], [17]. Recently, Srivastava revived the study of Hankel determinants, his pioneering work on the subject was followed by a huge flood of papers dealing with coefficient inequalities, Hankel determinants of order two and three of univalent and holomorphic functions. Different superordinating functions and their geometrical interpretations have motivated further research of the subject of geometric function theory. Functions like 4>(z) = Y+f, l+Bz, + z, ez, sin z, the Fibonacci sequence are some among them to quote. Our work was motivated by Babalola [3], Sharma [15], Vamshee Krishna [18], Ravichandran et al. [14], Srivastava et al. [16], [17] in general, and Sokol [4], [13] in particular, In this paper, we evaluate the bounds on a5, a6, a7, and H3(1) for f e S* (q). For f e S* (q), we estimate the bounds for Zalcman&s functional for n = 3, 4. Also, we define the class C (q) and make a similar study associated with shell-like curves for a function f e C (q).

Lemma 1. [12] If p eV is of the form p(z) = 1 + cYz + c2z2 + ..., then Kl< 2 V n e N.

Lemma 2. [9] For a Schwarz function w(z) = cYz + c2z2 + ..., and for any ^ e C we have

|c2 — ^c2| < max{1, |^|} .

Theorem 1. If f e S* (q) is of the form f (z) = z + aYz + a2z2 + ..., then

N < 13/24, |a6| < 29/30, |a71 < 309/288. Proof. As f e S*(q), by using subordination we get

= w(z) + y/1+ W2 (z).

zf (z) — f (z) w (z) = f (zW1+ W2 (z). (6)

Here w is the Schwarz function with the conditions w (0) = 0 and lw (z)l < 1 for |z| < 1, which can be represented as

w (z) = ^ cnzn, Vn e N with |cra| < 1. (7)

From relations (6) and (7) we have

VT+^M = 1 + f + (ClC^Z3 + (C1C3 + I2 - |)z4+

i CiC^W /Co 3 c2 C0 C1C3 Ci \\

+ C1C4 + C2C3 - J + + C1C5 + C2C4 - —p - -2- + 16 J ;

f (z)^1 + w2 (z) = z + «2Z2 + + ^z3 + ^CiC2 + ^ + a^)z4+

/ c2 ci «3С1 \\ 5 (

+ C1C3 + — - "8" + a,2CiC2 + -y + a,5)z + C1C4 + C2C33 2 4 2

C1C2 a2C2 , 0>2C1 a4C1 \\ 6

+ J — — - — - + - — - + - + - —--i_

V 2 4 2 16 2 2 2 8

+ C1C5 + C2C4 + CI2C1C4 + CI2C2C3 + Й4С1С2 + a,3C1C3 + a^jz1 + ... (8)

Further,

Z f&(z) - w(z)f(z) = Z + (2й2 - C1)Z2 + (3CI3 - (I2C1 - C2)z3 +

+ (4 С14 - CI3C1 - CI2C2 - C3)z4 + (5 a5 - CI4C1 - Ч3С2 - a2C3 - C4)z5+

+ (60,6 - CI5C1 - CI4C2 - CI3C3 - CI2C4 - C5)z6 +

+ (7ai - a,6C1 - CL5C2 - CI4C3 - CI3C4 - 0205 - C6)z1 + ... (9)

From (8) and (9), upon equating the coefficients of the same powers of z,

d2 = C1, (10)

«3 = T(c2 + 3 c1) , (11)

«4 = 3(4c3 + 2C1C2 + , (12)

a5 = 4 V3C1C3 + YC1C2 + 3C1 + C2 + °4) , (13)

d6 = 60 (27C1C4 + 20C2C3 + 19c3c2 + 20C1C2 + 30c^ + 3c5 + 12c^j , (14)

ai = 7^ Î400c3 + 1056C1C5 + 840C2C4 + 780c1c2 + 700c1c3+ 2880

+49c\\ + III6CÏC4 + 1720c 1C2C3 + 332c1c2 + 240c2 + 480c^. (15)

Raina and Sokol [13] estimated the bounds of the second, third, and fourth coefficients as |a2|< 1, |a3|< 3/4 and |a4|< 1/2. To estimate the bounds of the fifth, sixth, and seventh coefficients, we establish some properties of cn involved in (7). The function p (z) is given by

The Caratheodory function is defined by the property V,e{p(z)} > 0 in U, whose coefficients satisfy the condition

\\Pk | < 2 Vk G N. (17)

Equating the coefficients of the same powers of z in relation (16), we get

Pi = 2ci, (18)

P2 = 2 (CÏ + C2) , (19)

P3 = 2(cÏ + 2cic2 + C3 ), (20)

P4 = 2(c\\ + 3CÏC2 + 2ci c3 + c2 + C4), (21)

p5 = 2(c5 + 3cic2 + 3cfc3 + 4c3c2 + 2C2C3 + 2cic4 + c5), (22)

p6 = 2(c\\ + 5c4c2 + 4cÏc2 + 4c3c3 + c3, + c3 + 3ciC2C3 + 3cic4 + 2C2C4 + 2cic5 + c6)

p7 = 2(c[ + 3cic2 + 6c5c2 + 6cÏc2 + 4cic3 + 5cic3 + 9cic2c3 + 3c2c3+

+ 4c~l C4 + 3ciC2C4 + 2C3C4 + 3c\\c5 + 2c2C5 + 2cic6 + C7). (24)

Apply the condition in (17) to relations (18) to (24) and get

M < 1, (25)

I ci + C2 |< 1, (26)

I c3 + 2ci C2 + C3 |< 1, (27)

| ci + 3c2c2 + 2cic3 + c2 + C4 |< 1, (28)

| ci + 3cic2 + 3cfc3 + 4cÏC2 + 2C2C3 + 2cic4 + c51 < 1, (29)

| c6 + 5 c1c2 + 4c2c2 + 4 cfc3 + c2 + c2 + 3 C1C2C3 + 3 c^+

+2 c2 C4 + 2 C1C5 + ce|< 1

| c[ + 3 cic2 + 6 c1c2 + 6 c?c2 + 4cic2 + 5 c4c3 + 9 C2C2C3 + 3 c2c3 +

+ 4 c1c4 + 3 C1C2C4 + 2 C3C4 + 3 c1c5 + 2 C2C5 + 2 C1C6 + Ct|< 1.

From the relation (13), we have

— {14 C1C3 + 6 c2 + 4 ci + 17 C?C2 + 6 C4} ,

- ^ (+ C2)2 + ^C2 (c? + C2) - ^

By applying the triangle inequality to relation (32), we get

|«5 |< 24 |c4 + 3c1c2 + 2C1C3 + C2 + C4 | + — | (c! + C2) | +

+ ^21^2 (c1 + C2) | + 1241. (33)

We know that coefficients of the function w(z) satisfy |c^ < 1. By applying relations (26) and (28) to relation (33), we get

Using relation (14), we can estimate the bound on the sixth coefficient. 1

«6 = — 6 60

+ 7C1(cj + 3c1c2 + 2C1C3 + c2 + C4) - 7c2(c1 + 2C1C2 + C3) + + 14C3(c1 + C2) + 4c1c2 + 14C1C4 + 9C5I. (35)

Using the triangle inequality to relation (35), we get

|«6| < 3IC5 + 2C1C4 + 3c1c3 + 4c3C2 + 3C1C2 + 2C2C3 + C5I + 60

+ 7|ci||c4 + 3CÏC2 + 2CiC3 + c2 + C4|+7|Ci|2 jcf + 2CiC2 + C3I +

+ 14|c3||c2 + C2j+4jCijjC2j2+14jcijjC4j+9jC5j

We know that coefficients of the function w(z) satisfy |cra| < 1. By applying relations (26) to (29) to relation (36), it is reduced to

, , 29 M < 30.

Using relation (15), we can estimate the bound on the seventh coefficient

+ 2C2C4 + 2ci C5 + c6 ) + 50ci (c5 + 3cic2 + 3c2 C3 + 4cÏc2 + 2C2C3+ + 2CÏC4 + C5) + 40C2(4 + 3c2C2 + 2CÏC3 + c2 + C4) — 200c2(cÏ + 3cÏc2+ + 2CÏC3 + c2 + c4 ) + 200C3(4 + 2ciC2 + C3) — 50cÏ (ci + 2cic2 + £3) + + 400c4(cj + C2) + 540ciC2C3 — 90c2 c2(ci + a) + 50ci(cÏ + C2) —

— 168cic2 + 216CÏC4 + 280C6 + 606cic5

By applying the triangle inequality to relation (37) and using relations (26) to (30), we get

j«7j < 309.

j«2«3 — «4j < 1.

Proof. If f e S* (q) then, from Theorem 1, upon using the values of 0,2, a3, 04 from Equations (10), (11), and (12), obtain

Ci Ci / 2 \\ 1 / 3 \\

a,2a3 — aA = 3 + -3 (c^ + C2) — 3 (Ci + 2cic2 + c3) .

By applying the triangle inequality, ja2«3 — ^j <

f (d + =2)

From relations (25), (26), and (27), obtain

|«2«3 -ai|< 3 + 3 + 3 = L

l H (1) |< 265.

| 3V n< 192

Proof. Due to Raina and Sokol [13], |a21< 1, |a3|< 3/4, |a41< 1/2, |a2a4 — a2|< 39/48 and |a3 — a2|< 1/2. Substituting these bounds, relations (34) and (38) in the relation (4), we get

• , , i 3/3^ 1 , N 13 /1\\ |H (1)|< 4 (<48/ + 2 24 (2) &

Theorem 4. Let f-1 (z) = z + dn,zn be the inverse function of f.

For any ^ E C and f E S* (q) of the form f(z) = z+a1 z+a2z2+..., we get

| d3 — ^ d21 < 1 max <j 1,

Proof. By the definition of the inverse function, we have

/(/-1 (*)) = r1 ( /(*)) = z. (40)

Let f-1 (z) = z + ^2d,

(z) = z + > anz

From relations (1) and (40), it can be reduced to

/-1(z + a2z2 + a3Z3 + ...) = z. (41)

From (40) and (41) one obtains

z + Z2(^ci2 + d^j + z3(a,3 + 2a2<l2 + d^ +----= z

Comparing the coefficients of z2 and z3 on both sides, one can see that

«2 + d2 = 0, (42)

a,3 + 2a,2d2 + d3 = 0.

Now, from (12) and (42) we get

d2 = — Ci.

From (12), (13) and (43) we get

Now consider

d3 — ^ d2 = - < C2 — ci

{" — <5

d3 — ^d2 = - {c2 — v2 c2} , where v2 =

Applying Lemma 2 to relation (44), one gets estimate (39). This estimate is sharp, the equality is attained on the following functions

d3 — ^ d2

if p(z) if p(z)

Theorem 5. For a function f e S* (q) of the form f(z) = z + a2z + + a2z2 + ..., for any ^ e C, and for G (z) = jjf) = 1 + d2z + d2z2 + ..., we get

| d2 — ^ d21 < 1 max 11, Proof. As f eS * (q) and

Simplifying, one obtains that

—(j) = 1 — a2Z + (a2 — a^jz2 + ... (47)

From (46) and (47), upon equating the coefficients of the same powers of z, we have

d1 = — a2, (48)

From (10) and (48), we get d1 = — c1.

From (10), (11) and (49), we get d2 = — . Assuming that ^ is a complex parameter, take

d2 = a2 — a3. (49)

d2 — » d1 = 1 |c2 — <1 | .

d2 — ^ d\\ = 1 {c2 — V2 c!} , where V2 = —(50)

Applying Lemma 2 to relation (50), one gets (45), with equalities

№ — » d2| = ^ 2

if P(Z)- 1 — ^2&

•f ( ) 1+ *

lf p(Z) = --.

Theorem 6. If f E S * (q) is of the form f(z) = z + a1z + a2z2 + ... then

a3 — »a2\\ < ~2g3 maXi1, 2a 22 /

Proof. Let f E S* (q); then there exists a Schwarz function w such that

v & = w (z) + ^1 + w2(z). (52)

( f * 9) (z)

The left-hand side of (52) has the expansion

z((f *g)&(*)) ( 2

——-—-— = 1 + a2Q2Z + 2 (1393 — +

(f * 9) (z) V J

+ (3 0,494 — 3 0,20,39293 + a^g^z3 + ... (53)

Substituting the expansion w(z) = c2z + c2z2 + ... into the right-hand side of (52), one obtains

ci C2 3ci

Consider

a3 — ^ =2^1C2 — CA 922 )j ,

Applying Lemma 2 to relation (54), one gets (51). The equality arises for

if p(z) = +

, 2 | 53

Proof. If f e S* (q) then, from Theorem 1, upon using the values of a3, a5 from equations (11) and (13) one obtains that

^ — «5 = 7 ( C2 + - Ci) — J ^ C1C3 + 17 clc2 + 3 c2 + ^ + C^

a3 — 05 = - (c2 + C2) — — (c4 + 3c2C2 + 2C1C3 + c2 + q) +

+ 4 (=2 (cl + C2)) — ^ (C2 (c&2 + C2)) +16 —

By applying the triangle inequality, we have

a\\ — a5 <

+ C2)&

— (cl + 3cjc2 + 2C1C3 + cj + C4)

From relations (7), (25), (26), and (28) we get

+ +

I % - as I - 8 + 24 + 4 + 12 + 16 + 24 = 48 •

Thus the result is proved for the case n = 3 of the Zalcman conjecture for feS* (q). □

Theorem 8. If f e S* (q) is of the form f(z) = z + a\\z + a2z2 + •.., then

I a2 - •

Proof. If f e S* (q) then, from Theorem 1, upon using the values of a3, a5 from equations (12) and (15) one obtains that

az =9(4c? + 2cic2 + 2 — 2880 (4Q0c? + 1056C1C5 + 840C2C4+

+ 780clcj + 700clc 3 + 49 cf + 1116c1c4 + 1720cic2c3 + 332c1cj+

+ 240c? + 480c6).

az = 144( C1 + 2cic2 +c? )2 + 144c2 — 144( c? + 2cic2 +c?)( c?)+

+ 200( ci + 5 c4c2 + 4 cic2 + 4 c?c? + c2 + c2 + 3 cic2c? + 3 c2ci+ + 2 C2C4 + 2 cic5 + c6) + 40 C2 ( ci + 3 cic2 + 2 cic? + c2 + C4 ) —

— 200ci( ci + 3 c2c2 + +2 cic? + c2 + C4) + 200c?( c? + 2 dC2 + c?) —

— 50 ci( ci + 2 cic2 + c?) + 400C4( c2 + C2 ) + 540dC2C? — 90 c^ ci+

+ C2) + 50 c4( C2 + C2) —168 cic2 + 216 cic4 + 280Co + 606dC5

By using the triangle inequality and relations (7) and (25) - (31) to relation (55), we get the required result. Thus, the result is proved for the case n = 4 of the Zalcman conjecture for f G S * (q). □

Theorem 9. If f G C (q) is of the form f(z) = z + a2z + a2z2 + • • •, then

M < 1/2, M < 1/4, M < 7/24, H < 3/40.

Proof. If the function f G C (q), then from relation (5) we have

f (z) + z f (z) - f (z) w (z) = f&(zWl+w2 (z). (56)

From the relations (7) and (56) we have

f (z) + w2 (z) = 1 + 2d2Z + (3a3 + ^z2 + (4a4 + (^2 + cid) z3+ + ^5a5 + 2CI2C1C2 + c 1C3 + ^ + 7° - + (6 a6 + 2c\\aA+

+ c 1C4 + 3 a3CiC2 + 2(Ï2C1C3 + a2cl + C2C3 - - + ... (57)

Further,

f (z) + z f" (z) - f (z) w (z) = 1 + (4ai - d)z + (9a3 - 2aiC 2 - d) z2+ + (16a4 - 3a3c 1 - 2a2c2 - c3)z3 + (25a5 - 4a4c 1 - 3a3c2 - 2a2c3 - c4)z4+ + (36a6 - 5 a5c 1 - 4a4c2 - 3a3c3 - 2a2c4 - c5)z5 + ... (58)

From (57) and (58), upon comparing the coefficients of the same powers of , we get

= f, (59)

= 1 ("2 + f )• (60)

^ — ^ ^ (61)

a2 a GÛ + ^ 6

+ Ç1C3 + Ç2 + 12 20 20,

a5 = 120 ^C4 + 3c2c2 + 2°ic3 + c2 + c4) + (c2 + c2)2 - 2c4 + C4) . (62)

By applying the triangle inequality to relations (59)-(62), also using relations (26)-(28), we get the required result.

Here the coefficients a2 and a3 have the best bounds for the function fi(z), which is defined as

then f[(z) + z f{(z) - zf[(z) = f[(z)VTT^.

After simplification, we get b2 = 1/2, b3 = 1/4. □

Theorem 10. If f E C (q) is of the form f(z) = z + a\\z + a2z2 + ..., then

\\a2a3 - a4\\< 29/48

Proof. If f E C (q), then, from Theorem 9 and upon using the values of a2, a3, a4 from equations (59), (60), and (61) one obtains

CL2d3 — 0,4

— "24" ($ + + ^ (4 + 2 ClC2 + C3) .

By applying the triangle inequality, it is reduced to

(c2 + + 12 (c3 +2 Clc2 + ^

From relations (25), (26), and (27) we obtain that

. 11 7 1 29

^ — a4|< 48 + 24 + 12 = 48.

| a3 — ^câ l< -ma^i 1 6 I

where ^ is a complex number and the best bound is obtained.

Proof. If f E C (q), then, from Theorem 9 and upon using the values of a2, a3, from equations (59) and (60) one obtains that

( C2 — UiC^j

where v\\ = 1 ^J. By applying Lemma 2 to Equation (64), one obtains the result as in Equation (63). The sharpness is given below:

f f ) 1 + z

if pfz) = --.

Theorem 12. If f E C (q) is of the form f(z) = z + a2z + a2z2 + ..., then

I«2«4 - «2|< 31/144.

Proof. If f E C (q), then from Theorem 9 and upon using the values of «2, «3, a4, from equations (59), (60) and (61) one obtains that

a2a4 — a3 =

d f 4 + 2 d d + c3) + -12 cfd + 9c id + 8 c22

By applying the triangle inequality to relation (65) and also using relations (25) to (27), we get the required result. □

Theorem 13. If f E C fq) is of the form ffz) = z + a\\Z + a2z2 + ..., we have

Proof. Substituting the results of Theorems 9, 10, 11, and 12 in the relation (4), we get

lumi .1 / 19 \\ 7 f 29\\ 3 f 1 \\ 1277

\\H3 f1)\\ < 4[m)+û\\4E) + 400 \\6) = 5760 ■

Theorem 14. If f E C fq) is of the form ffz) = z + a\\z + a2z2 + ■ ■ ■, then

I a2 - a5 I < ^.

Proof. If f E C (q), then, from Theorem 9 and upon using the values of a3, a5, from equations (60) and (62) one obtains that

a3 -a* = 72° (2°(c2 + c2 )2 - 3°(c4 + 3 c2c2 + 2ci C3 + c2 + 04) +

+ 2°c?(C? + C2) - 12C2(c2 + C2) - 6C4 + 11 c4 + 6c2) By applying the triangle inequality to relation (66), we have

\\a3 - a*\\ < 722° (2°\\c? + c2\\2+3°\\c\\ + 3cfe + 2C1C3 + c2 + C4\\ +

+ 2°\\Cl\\2\\c2 + C2\\ + 12\\C2\\\\C? + C2\\+6\\c4\\ + 11\\Cl\\4+6\\C2\\^ .

From relations (7), (25), (26), and (28), we get

I «2 «51 < 720 + 720 + 720 + 720 + 720 + 720 + 720 48&

Acknowledgement. The authors are very much thankful to the Editor and the esteemed referees for their careful reading and valuable suggestions, which improved the paper.

The work presented in this paper is partially supported by DST/FIST grant no: SR/FST/MSI-101/2014, dated:14-01-2016.

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Received September 07, 2019. In revised form, February 28, 2020. Accepted March 03, 2020. Published online March 27, 2020.

V. Suman Kumar

Department of Mathematics, TSMS, Chigurumamidi, Karimnagar,

Telangana-505481, India

E-mail: suman.vodnala@gmail.com

R. B. Sharma

Department of Mathematics, Kakatiya University, Warangal, Telangana-506009, India E-mail: rbsharma005@gmail.com