SHARP BOUNDS ASSOCIATED WITH THE ZALCMAN CONJECTURE FOR THE INITIAL COEFFICIENTS AND SECOND HANKEL DETERMINANTS FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Апрель—июнь, 2023. Том 30, № 2

UDC 517.54

SHARP BOUNDS ASSOCIATED WITH THE ZALCMAN CONJECTURE FOR THE

INITIAL COEFFICIENTS AND SECOND HANKEL DETERMINANTS FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS N. Vani, D. Vamshee Krishna, and B. Rath

Abstract: In this paper, we obtain sharp bounds in the Zalcman conjecture for the initial coefficients, the second Hankel determinant H2,2(f) = «2«4 — a§ and an upper bound for the second Hankel determinant H23 (f ) = «305 —<2 for the functions belonging to a certain subclass of analytic functions. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Caratheodory class P.

DOI: 10.25587/SVFU.2023.24.67.007 Keywords: analytic function, upper bound, the Zalcman conjecture, univalent function, Caratheodory function.

1. Introduction

Let H denote the class of all analytic functions defined in the open unit disc D = {z G C : |z| < 1}. Let A represent the class of functions f G H satisfying the normalized conditions namely f (0) = f'(0) — 1 = 0, i.e., of the form

f (z) = YI anzn, a1 := 1, z G D.

(1.1)

n=1

By S, we denote the subfamily of A, consisting of all univalent functions (i.e., one-to-one) in D. Pommerenke [1] characterized the nth Hankel determinant of order r, for f given in (1.1) with r, n G N = {1,2, 3,...} as

Hr,n(f ) =

an an+i an+i an+2

an+r-1 an+r

an+r-1 an+r

an+2r-2

(1.2)

The Fekete-Szego functional is obtained for r = 2 and n = 1 in (1.2) and denoted by #2,1 (f), where

#2,1 (f ) =

a1 a2 a,2 аз

= a3 — a2.

© 2023 N. Vani, D. Vamshee Krishna, B. Rath

Further, sharp bounds for the functional |a2a4 — a2| are obtained in (1.2) for r = 2 and n = 2, the Hankel determinant of order two

#2,2(7 )

ß2 03

ß3 ß4

2

ß2®4 — ®3-

In recent years, many authors have focussed research on the estimation of an upper bound for |#2,2(7)|. The exact estimates of |#2,2(7)| for the family of univalent functions, namely bounded turning, starlike and convex, denoted by S* and K, respectively, fulfilling the analytic conditions Re{7'(z)} > 0, Re{ } > 0 and

Re{l + } > 0 in the unit disc D, were proved by Janteng et al. (see [2,3]), the bounds as 4/9, 1, and 1/8 were derived. For recent results on the second Hankel determinants (see [4-8]). Similarly, by taking r = 2 and n = 3 in (1.2), we have #2,3(f) = a3a5 — a2, the second Hankel determinant, for which Zaprawa [9] derived sharp bounds |#2,3 (7) I < 1 f°r the class S* and |#2,3(7) I < is class with

the assumption that a2 = 0 in f given in (1.1). By the results derived by Zaprawa [9], recently, Andy Liew Pik Hern et al. [10] have shown that |#2,3(7)l < if f°r 7 G S* and |#2,3(7)| < ^ for 7 G J^s, where S* and denote the families of

16

__-p O. Q* onrl 'y?^ rl /21VI /"l f /21 f Vl /21 fom'

240

starlike and convex functions with respect to symmetric points, analytically defined

as

,* ^ I 2f '(z)

f2(\znf(' ,}>0, ZGD. (1.4)

lzf'(z) + zf'(—z) J

Choosing r = 2 and n = p + 1 in (1.2), we obtain the Hankel determinant of second order for the p-valent function (see [11])

#2,p+l(7) —

ßp+1 ßp+2 ap+2 ap+3

2

— ap+1ap+3 — ap+2;

In the 1960s Zalcman posed a conjecture that if f € S then

|a;; — fl2n-i| < (n — 1)2 for n = 2, 3,...; (1.5)

the equality holds only for the Koebe function k(z) = z/(1 — z)2 or its rotations. For functions in S, Krushkal proved the Zalcman conjecture for n = 3 (see [12]) and recently for n = 4,5, 6,... [13]. This remarkable conjecture was investigated by many researchers and is still an open problem for functions belonging to class S when n > 6. The Zalcman conjecture was proved for certain special subclasses of S, such as starlike, typically real, and close-to-convex functions (see [12,14]). Recently, Abu Muhanna et al. [15] solved the Zalcman conjecture for the class F consisting of the functions f € A satisfying the analytic condition

Re (l + j^j > -1/2, z G D.

Functions in the class F are known to be convex in some direction (and hence close-to-convex and univalent) in D. In 1988, Ma [16] proved the Zalcman conjecture

for close-to-convex functions. For f G S, Ma [17] proposed a generalized Zalcman conjecture:

— an+m-1| < (n — 1)(m — 1) for to, n = 2, 3,..., (1.6)

which is still an open problem, and proved it for classes S* and SR, where SR denotes the type of all functions in A which are typically real. Bansal and Sokol [18] studied the Zalcman conjecture for some subclasses of analytic functions. Ravichandran and Verma [19] proved this conjecture for the classes of starlike and convex functions of a certain order and the class of functions with bounded turning. Motivated by the results mentioned above, which are associated with the Zalcman conjecture and the Hankel determinants, in the present paper, we are attempting to find sharp upper bounds for the coefficient inequalities specified in the abstract for the functions belonging to a certain subclass of analytic functions defined as follows.

Definition [20]. A mapping f G A is said to be in the class S*KS(P) (0 < P < 1) if

Re

2{ zf '(z) + Pz2f "(z)}

(1 — P) {f (z) — f (—z)} + P {zf'(z) + zf'(-z)}

> 0, z G D. (1.7)

For P = 0 and P = 1 in (1.7), we get S*KS(0) = S*, consisting of starlike functions with respect to symmetric points, interpreted and studied by Sakaguchi [21], and S*K(1) = K, consisting of convex functions with respect to symmetric points, analyzed by Das and Singh [22], for which analytic conditions are given in (1.3) and

(1.4).

In proving our results, the required sharp estimates specified below are given as lemmas suitable for functions possessing positive real part.

Let P be a class of all functions g having a positive real part in D:

g(z) = 1+£ ctzt, (1.8)

,tzt t=i

Every such a function is called Caratheodory function [23].

Lemma 1.1 [24]. If g G P, then |ct| < 2 for t G N; the equality is attained for the function h(z) = j^, z G D.

Lemma 1.2 [25]. If g G P, then the estimate — ^Cjci-j | < 2 holds for G N = {1,2, 3,...} with i > j and ^ G [0,1]. From Lemma 1.2, Livingston [26] proved that |c^ — Cjci-j-1 < 2.

Lemma 1.3 [9]. If g G P, then |c2c4 — c§| < 4. The inequality holds only for the functions

1 + z2 1 + z 3

and their rotations.

Lemma 1.4 [27]. Let g G P be of the form (1.8) with c? > 0. Then

2c2 = c? + y(4 - c?)

and

4c3 = [c? + {2c?y - c?y2 + 2(1 - |x|2)y}(4 - c?)] , for some complex valued x and y such that |x| < 1 and |y| < 1.

To obtain our results, we adopt some ideas from Libera and Zlotkiewicz [27].

2. Important Results Theorem 2.1. If f G S*K(£) (0 < £ < 1), then

\a2a3 - 041 < 2{ll+W) <{2~ 1)(3 " 1} = 2;

this inequality is sharp for g?(z)

l±2l

1-z3

Proof. For f G S*K(£), there exists g G P such that 2{zf '(z )+ £z2f ''(z)}

= ). (2.1)

(1 - £){f (z) - f (-z)} + £{zf'(z) + zf'(-z)} Putting the values for f, f' , f'' and g in (2.1), we get

[2(1 + £)a? + 3(1 + 2£)a?z + 4(1 + 3£)a4z2 + 5(1 + 4£)asz3 + ... ] = [c? + {c? + (1 + 2£)a?}z + {c? + (1 + 2£)cia?}z2

+ {c4 + (1 + 2£)c2a? + (1 + 4£)as}z3 + ... ]. (2.2)

Equating the coefficients for powers of z in (2.2), we obtain

_ Cl _ C2 _ (2c3 + c1c2) _ (2c4 + c|)

a2-2(TT^' 2(1 + 2/3)' 8(1 + 3/3) ' as~ 8(1 + 4/3) '

Using the values of a2, a3 and a4 from (2.3), we have

c?c2 (2c3 + c? c2)

a2a3 — a4 —

4(1+ ß )(1 + 2ß) 8(1+3ß )

1 ( (—2ß2 + 3ß +1)

c3 - , „w, , 1C2

4(1 + 3P) \ 3 2(1+ P)(1+2P)

Taking modulus on both sides and then applying Lemma 1.2 to the expression above, upon simplification, we obtain

|a2a3 - o4| < 2(1 | < (2 - 1)(3 - 1) = 2. □

Remark 2.2. For the extremal function gi(z) = jzfs = 1 + 2z3 + 2z6 + ..., we have c1 = 0, c2 = 0, and c3 = 2. Hence, from (2.3) we obtain a2 = 0, a3 = 0,

and a4 = 4(1+3)g) •

Theorem 2.3. If f € S*Ks(£) (0 < £ < 1), then

|a2-a3|<^<(2-l)2 = l;

2

la-

this inequality is sharp for <72(2) =

Proof. Using the values of a2 and a3 from (2.3), we have

2_ =_£i___c2 = 1 / (1 + 2£) 2\

&2 &3 4(1+£)2 2(1 + 2£) 2(1 + 2/3) \ 2 2(1+^)2CV-

Putting modulus on both sides in the expression above and applying Lemma 1.2, after simplifying, we get

|a2"a31-(TT^)- D

Remark 2.4. For the extremal function <72(2) = = 1 + 2z2 + 2z4 + ..., we have c\ = 0 and c2 = 2; Hence, from (2.3), we obtain a2 = 0 and a3 = 2(1+2/3) •

Theorem 2.5. If f € S*Ks(£) (0 < £ < 1), then

this inequality is sharp for gz{z) = jz^z-

Proof. Using the values of a3 and a5 from (2.3), we have

_ 4 (2C4+4)_ 1 f (—4/32 +4/3+1) 2\

3 5 ~ 4(1 + 2£)2 8(1 + 4/3) ~~ 4(1 + 4/3) \ 2(1+2£)2

Taking modulus on both sides and applying Lemma 1.2, after simplifying, we get

I a? — as I < -r • D

13 5| " 2(1+4/3)

Remark 2.6. For the extremal function

1 + z4

ff3(z) = ^ = l+2z4 + 2z8 + ...,

1 — z4

we have c2 = 0 and C4 = 2, therefore, from (2.3), we obtain a3 = 0 and as = ■

Theorem 2.7. If f € S*Ks(£) (0 < £ < 1), then

1

|#2,2(f)| = |a2«4 — a2| <

(1 + 2£)2'

the inequality is sharp for the same function g2(z) as in Theorem 2.3.

Proof. Using the values of a2, a3, and a4 from (2.3), for the expression a2a4 — a2, we get

1

2

a2a4 — a3

16(1 + £)(1 +2£ )2(1 +3£)

0/,A2c1c2 — 4(1 + £ )(1+3£)c2

x (2(1 + 2£)2C1C3 + (1 + 2£)2c1c2 — 4(1 + £)(1 + 3£)c2), (2.5)

which is equivalent to

a2Q4 " = 16(1+P)(1 + 2P)2(1+3P) [dlClC3 + d2Ci°2 + ^ ' (2'6)

where

di = 2(1 + 2^)2, d2 = (1+2P)2, d3 = -4(1+ P )(1 + 3P). (2.7)

Putting the values of c2 and c3 from Lemma 1.4 into the right-hand side of (2.6), we simplify it into

4 [dicic3 + d2c2c2 + d3 c2] = [(di + 2d2 + d3)ci

+ 2(di + d2 + d3)c2(4 - ci)x - dic2(4 - c2)x2 + d3(4 - ci)2x2 +

2dici(4 - c2)(1 - |x|2)y]. (2.8)

Taking modulus on both sides and applying the triangle inequality in the expression above, we get

4|dicic3 + d2ci c2 + d3c2| < [|di +2d2 + d3||ci|4 +2|di||ci||4 - c2||y|

+ 2|di + d2 + d3||ci|2|4 - c2||x| + {(|di |- |d3|)c2 - 2|di||ci||y| +4|d3|} |4 - c2||x|2].

(2.9)

By (2.7), we can now write

|di +2d2 + d3| = 4P2, |di + d2 + d3| = 1 + 4p, (2.10)

{(|di|-|d3|)c2 - 2|di||ci||y| +4|d3|}

= -(4p2 + 8P + 2)c2 - 4(1 + 2P)2ci|y| + 16(1 + P)(1 + 3P) = 2(ci - 2){-(2p2 + 4P + 1)ci - 4(1 + p)(1 + 3P)},

= 2(2 - ci){(2p2 + 4P + 1)ci + 4(1 + P)(1 + 3P)}, |y| = 1.

Putting the calculated values from (2.10) and the value of di from (2.7) into (2.9), after simplifying, we get

2|dicic3 + d2c2c2 + d3c21 < [2p2ci + 2(1 + 2P)2ci(4 - c?)|y| + (1 + 4P)c?(4 - c?) |x| + (2 - ci){(2P2 + 4P + 1)ci + 4(1 + P)(1 + 3P)}(4 - c2) |x|2]. (2.11)

Applying the triangle inequality, restoring |x| by p, with |y| < 1, choosing c2 = c e [0, 2], on the right-hand side of (2.11) we obtain

2|dicic3 + d2cic2 + d3c2| < [2P2c4 + 2(1 + 2P)2c(4 - c2) + (1 + 4P)c2(4 - c2)p

+ (2-c){(2P2 +4P + 1)c+4(1+ P)(1 + 3P)}(4-c2)p2] = H(c,p) for |x| = p e [0,1].

(2.12)

Here

H(c, p) = [2P2c4 + 2(1 + 2P)2c(4 - c2) + (1 + 4P)c2(4 - c2)p

+ (2 - c){(2P2 + 4P + 1)c + 4(1 + P)(1 + 3P)}(4 - c2)p2]. (2.13)

To determine the maximum value of H(c, p) over the rectangle [0,1] x [0, 2], we consider the partial differential coefficient of H(c, p) from (2.13) with regard to p given by d H

— = [(1 + 4/3)c2 + 2(2 — c){(2/32 + 4/3 + l)c + 4(1 + /3)(1 + 3/3)}p](4 — c2). (2.14) dp

For p G (0,1), c G (0,2), and (0 < /3 < 1), by (2.14), we notice that ff > 0, which indicates that H(c, p) turns out to be an increasing mapping of p, hence, its maximum value is attained on the boundary of the rectangle only, i.e., when p = 1. Therefore, for p =1 in (2.13), after simplifying, we get

F (c) = H (c, 1) = 4£2c4 - 8(1 + 2£)2c2 + 32(1 + £)(1 + 3£), (2.15)

F'(c) = 16£2c3 - 16(1 + 2£)2c, (2.16)

F''(c) = 48£2c2 - 16(1+ 2£)2. (2.17)

For the extreme values of F(c), let F'(c) = 0. From (2.16), we have

16c{£2c2 - (1+2£)2} = 0. (2.18)

Now, let us discuss the following two instances.

Case 1. When c =0, from (2.17), we note that

F''(0) = -16(1+2£)2 < 0 for 0 < £ < 1.

Therefore, by the 2nd differentiation test at c = 0, F(c) possesses the maximum value, which we can obtain from (2.15) as

max F(0) = 32(1 + £)(1+3£). (2.19)

Case 2. When c =0, from (2.18), we get

2 (1 + 2£)

(2.20)

£2

For 0 < £ < 1, from (2.20) we note that c2 does not belong to [0,2]. Now, simplifying the expressions (2.12) and (2.19), we obtain

|dicic3 + d2c1c2 + d3c2| < 16(1 + £)(1 + 3£). (2.21)

From (2.5) and (2.21), after simplifying, we get

ha4-a||< (1+12/3)2- □ (2-22)

Remark 2.8. For the extremal function <72(2) = ^rf? = 1 + 2z2 + 2z4 + ..., we have c1 =0, c2 = 2, c3 = 0, and c4 = 2, for which from (2.3) we obtain a2 = 0, 0,3 = 2(1+2/3)' and a4 = 0-

Remark 2.9. For £ = 0 and £ = 1 in (2.22), the particular results coincide with that of Rami Reddy and Vamshee Krishna [28].

Theorem 2.10. If f G S*Ks(P) (0 < £ < 1), then

13

|#2,3(/)| = - aj\ <

16(1 + 2£)(1 +4£)'

Proof. Using the values of a3, a4, and a5 from (2.3) in a3a5 — a|, we simplify it into

(4c3 + 8C2C4) (4C1C2C3 +4c3 + cfc|)"

2_ 1

a3a 5 — a4 — —

64

_(1 + 2£)(1+4£) (1 + 3£)2

Rearranging the terms in (2.23), we have

.J (1 + 2/3)(1 + 4/3) 2 4 \C2C4 - 4(1 + 3/3)2 C3

(2.23)

a3a5 — a4 —

.2 1

64(1 + 2£)(1 +4£)

}

/ (1 + 2/3) (1 + 4/3) + 4CHC4- 4(1 + 3/3)2 ClC3

cj f (1 + 2/3)(1 + 4/3) 2) 3c|

+ , „„w, , S 2--, 0^2-C1 f +

(1+2£)(1+4£) L 2 (1 + 3£)2 XJ (1 + 2£)(1+4£)

}

}

. (2.24)

Taking modulus on both sides and applying Lemmas 1.1, 1.2, and 1.3, upon simplification, we obtain

13

|#2,3(/)| = |o-3Q-5 ~ Q41 < 16(1 + 2/3)(1 + 4/3)' D (2'25)

Remark 2.11. For p = 0 and P = 1 in (2.25), the results coincide with that of Andy Liew Pik Hern et al. [10].

Acknowledgement: The authors would like to thank the esteemed referee(s) for their careful readings, valuable suggestions, and comments which helped improve the presentation of the paper.

REFERENCES

1. Pommerenke Ch., "On the coefficients and Hankel determinants of univalent functions," J. Lond. Math. Soc., 41, 111-122 (1966).

2. Janteng A., Halim S. A. and Darus M., "Hankel determinant for starlike and convex functions," Int. J. Math. Anal., 1, No. 13, 619-625 (2007).

3. Janteng A., Halim S. A. and Darus M., "Coefficient inequality for a function whose derivative has a positive real part," J. Inequal. Pure Appl. Math., 7, No. 2, 1-5 (2006).

4. Gurusamy P. and Jayasankar R., "The estimates for second Hankel determinant of Ma-Minda starlike and convex functions," AIP Conf. Proc., 2282, 020039 (2020).

5. Orhan H., Magesh N., and Balaji V. K., "Second Hankel determinant for certain class of bi-univalent functions defined by Chebyshev polynomials," Asian-Eur. J. Math., 12, No. 2, Article ID 1950017 (2019).

6. Sim Y. J., Thomas D. K., and Zaprawa P., "The second Hankel determinant for starlike and convex functions of order alpha," Complex Variables, Elliptic Equ., 67, No. 10, 2423-2443 (2021).

7. Sokol J. and Thomas D. K., "The second Hankel determinant for alpha-convex functions," Lith. Math. J., 58, No. 2, 212-218 (2018).

8. Zaprawa P., "Second Hankel determinants for the class of typically real functions," Abstr. Appl. Anal., 2016, Article ID 3792367 (2016).

9. Zaprawa P., "On Hankel determinant H2(3) for univalent functions," Result. Math., 73, No. 3 (2018).

10. Hern A. L. P., Janteng A., Omar R., "Hankel determinant H2(3) for certain subclasses of univalent functions," Math. Stat., 8, No. 5, 566-569 (2020).

11. Vamshee Krishna D. and RamReddy T., "Coefficient inequality for certain p-valent analytic functions," Rocky Mt. J. Math., 44, No. 6, 1941-1959 (2014).

12. Krushkal S. L., "Univalent functions and holomorphic motions," J. Anal. Math., 66, 253-275 (1995).

13. Krushkal S. L., "Proof of the Zalcman conjecture for initial coefficients," Georgian Math. J., 17, 663-681 (2010).

14. Brown J. E. and Tsao A., "Proof of the Zalcman conjecture for starlike and typically real functions," Math. Z., 191, 467-474 (1986).

15. Abu Muhanna Y., Li L., and Ponnusamy S., "Extremal problems on the class of convex functions of order -1/2," Arch. Math. (Basel), 103, No. 6, 461-471 (2014).

16. Ma W., "The Zalcman conjecture for close-to-convex functions," Proc. Amer. Math. Soc., 104, 741-744 (1988).

17. Ma W., "Generalized Zalcman conjecture for starlike and typically real functions," J. Math. AnaL. Appl., 234, 328-339 (1999).

18. Bansal D. and Sokol J., "Zalcman conjecture for some subclass of analytic functions," J. Fractional Calc., 8, No. 1, 1-5 (2017).

19. Ravichandran V. and Verma S., "Generalized Zalcman conjecture for some classes of analytic functions," J. Math. Anal. Appl., 450, No. 1, 592-605 (2017).

20. Vamshee Krishna D. and Shalini D., "Certain coefficient inequalities associated with Hankel determinant for a specific subfamily of holomorphic mappings," Publ. Inst. Math. (Beograd) (N. S.), 111, No. 125, 127-132 (2022).

21. Sakaguchi K., "On a certain univalent mapping," J. Math. Soc. Japan, 11, 72-75 (1959).

22. Das R. N. and Singh P., "On subclass of schlicht mappings," Ind. J. Pure Appl. Math., 8, 864-872 (1977).

23. Duren P. L., Univalent Functions, Springer, New York (1983) (Grundlehren Math. Wiss.; vol. 259).

24. Pommerenke Ch., Univalent Functions, Vandenhoeck und Ruprecht, Gottingen (1975).

25. Hayami T. and Owa S., "Generalized Hankel determinant for certain classes," Int. J. Math. Anal., 4, 2573-2585 (2010).

26. Livingston A. E., "The coefficients of multivalent close-to-convex functions," Proc. Amer. Math. Soc., 21, No. 3, 545-552 (1969).

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

28. RamReddy T. and Vamshee Krishna D., "Hankel determinant for starlike and convex functions with respect to symmetric points," J. Ind. Math. Soc. (N. S.), 79, No. 1-4, 161-171 (2012).

Submitted February 22, 2023 Revised May 11, 2023 Accepted May 29, 2023

N. Vani, D. Vamshee Krishna, and B. Rath Department of Mathematics GITAM School of Science, GITAM University, Visakhapatnam-530 045, A. P., India

[email protected], [email protected] (corresponding author), [email protected]