on ozaki close-to-convex functions

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Let S be the subclass of A consisting of univalent (that is, one-to-one) functions. A function f 2 A is called starlike (with respect to the origin) if f(D) is starlike.
Bull. Aust. Math. Soc. 99 (2019), 89–100 doi:10.1017/S0004972718000989

ON OZAKI CLOSE-TO-CONVEX FUNCTIONS VASUDEVARAO ALLU , DEREK K. THOMAS and NIKOLA TUNESKI (Received 4 July 2018; accepted 30 July 2018; first published online 20 September 2018) Abstract P n Let f be analytic in D = {z 2 C : |z| < 1} and given by f (z) = z + 1 n=2 an z . We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates. 2010 Mathematics subject classification: primary 30C45; secondary 30C55. Keywords and phrases: analytic, univalent, strongly close-to-convex functions, coefficient estimates.

1. Introduction and definitions Let A denote the class of functions f analytic in the unit disc D := {z 2 C : |z| < 1} with Taylor series 1 X f (z) = z + an zn . (1.1) n=2

Let S be the subclass of A consisting of univalent (that is, one-to-one) functions. A function f 2 A is called starlike (with respect to the origin) if f (D) is starlike with respect to the origin and convex if f (D) is convex. Let S⇤ (↵) and C(↵) denote respectively the classes of starlike and convex functions of order ↵ for 0  ↵ < 1 in S. It is well known that a function f 2 A belongs to S⇤ (↵) if and only if Re (z f 0 (z)/ f (z)) > ↵ for z 2 D, and f 2 C(↵) if and only if Re(1 + z f 00 (z)/ f 0 (z)) > ↵. Similarly, a function f 2 A belongs to K, the class of close-to-convex functions, if and only if there exists g 2 S⇤ such that Re [ei⌧ (z f 0 (z)/g(z))] > 0 for z 2 D and ⌧ 2 ( ⇡/2, ⇡/2). Thus, C ⇢ S⇤ ⇢ K ⇢ S. When ⌧ = 0, the resulting subclass of close-to-convex functions is denoted by K0 . Although the class K was first formally introduced by Kaplan [5] in 1952, already in 1941 Ozaki [9] considered functions in A satisfying the condition ✓ z f 00 (z) ◆ 1 Re 1 + 0 > (z 2 D). (1.2) f (z) 2

It follows from the original definition of Kaplan [5] that functions satisfying (1.2) are close-to-convex and therefore members of S. c 2018 Australian Mathematical Publishing Association Inc. 89

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Kargar and Ebadian [6] considered the following generalisation to (1.2). Definition 1.1. Let f 2 A be locally univalent for z 2 D and let 1/2 <  1. Then f 2 F ( ) if and only if ✓ z f 00 (z) ◆ 1 Re 1 + 0 > (z 2 D). (1.3) f (z) 2 Clearly, when 1/2 <  1/2, functions defined by (1.3) provide a subset of C, with F (1/2) = C, and, since 1/2 1/2 when  1, functions in F ( ) are closeto-convex when 1/2   1. We shall call members of f 2 F ( ) when 1/2   1 Ozaki close-to-convex functions and denote this class by FO ( ). For 0 <  1, the classes S⇤⇤ ( ) of strongly starlike functions and C⇤⇤ ( ) of strongly convex functions are defined for f 2 A and z 2 D, respectively, by z f 0 (z) ⇡ arg < f (z) 2 and ✓ z f 00 (z) ◆ ⇡ arg 1 + 0 < . f (z) 2 Functions in S⇤⇤ ( ) and C⇤⇤ ( ) are more difficult to deal with than those in S⇤ and C, and relatively few exact coefficient bounds are known. Sharp bounds are known only for functionals involving the coefficients a2 , a3 and a4 (see [1–3] and [17]). Even more elusive are sharp bounds for the class K ⇤⇤ ( ) of strongly close-to-convex functions, defined for f 2 A and z 2 D, by z f 0 (z) ⇡ arg < , g(z) 2 where 0 <  1 and g 2 S⇤ . It is a relatively simple exercise to obtain sharp bounds for the coefficients |a2 | and |a3 | when f 2 K ⇤⇤ ( ), but finding sharp bounds for |a4 | appears to be a more difficult problem. We note that in contrast to the definition of K, the definition of F ( ) does not involve an independent starlike function g, but, as was shown in [11], members of F (1) have coefficients which grow at the same rate as those in K, that is, O(n) as n ! 1. We make the following definition, which extends (1.3), the special case with = 1. Definition 1.2. Let f 2 A for z 2 D, with 0 <  1 and 1/2   1. Then f is called strongly Ozaki close-to-convex if and only if ✓2 1 2 ✓ z f 00 (z) ◆◆ ⇡ arg + 1+ 0 < (z 2 D). (1.4) 2 +1 2 +1 f (z) 2 We denote this class of functions by FO ( , ).

The primary object of this paper is to obtain sharp bounds for the coefficients |a2 |, |a3 | and |a4 |, and the corresponding inverse coefficients, for strongly Ozaki close-toconvex functions, thus providing sharp inequalities for the fourth coefficient of a class of strongly close-to-convex functions. We also give some distortion theorems.

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2. Lemmas We will use the following lemmas (see, for example, [1]) for functions p 2 P, the class of functions with positive real part in D, given by p(z) = 1 +

1 X

pn zn .

n=1

Lemma 2.1. If p 2 P, then |pn |  2 for n p2

µ 2 p  max{2, 2|µ 2 1

Also, |p2

1 2 2 p1 |

1 and ( 2, 1|} = 2|µ 2

Lemma 2.2. Let p 2 P. If 0  B  1 and B(2B |p3

Lemma 2.3. If p 2 P, then |p3

0  µ  2, 1|, elsewhere.

1 2 2 |p1 |.

1)  D  B, then

2Bp1 p2 + Dp31 |  2.

(µ + 1)p1 p2 + µp31 |  max{2, 2|2µ

1|} =

(

2, 2|2µ

0  µ  1, 1|, elsewhere.

We will also use the following result from the theory of di↵erential subordination (see [8]). Lemma 2.4. Let ⌦ ⇢ C and suppose that the function : C2 ⇥ D ! C satisfies (ix, y; z) < ⌦ for all x 2 R, y  n(1 + x2 )/2 and z 2 D. If p is analytic in D, p(0) = 1 and (p(z), zp0 (z); z) 2 ⌦ for all z 2 D, then Re p(z) > 0 for z 2 D.

The following result (see [12] and [4, page 67]) is often useful and we will need it in Theorem 3.4. Lemma 2.5. Suppose that f 2 S and that z = rei✓ 2 D. If m0 (r)  | f 0 (z)|  M 0 (r),

where m0 (r) and M 0 (r) are real-valued functions of r in [0, 1), then Z r Z r m0 (t) dt  | f (z)|  M 0 (r) dt. 0

0

Although functions in F ( ) are close-to-convex when 1/2   1, Ponnusamy et al. [11] gave an example to show that when = 1, they are not necessarily starlike. On the other hand, we will show in this paper that when the second coefficient of the Taylor expansion for f (z) is zero, functions in F (1) are starlike of order 1/2, that is, Re(z f 0 (z)/ f (z)) > 1/2. In the next section, we consider the class F ( ), that is, when 1/2   1. The following sections will be concerned with Ozaki close-to-convex functions, that is, when 1/2   1.

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3. The class F ( )

Theorem 3.1. Let An be the set of functions in A given by If f 2 F ( ) for where

f (z) = z + an+1 zn+1 + an+2 zn+2 + · · · . 1/2   1, 0  ↵ < 1, n 2 N and b = (↵, n) = min{ ⇤ (↵, n), 1},

81 > > > > > > > 1 > > : 2 ⇤ b then An \ F ( ) ⇢ S (↵).

n 2 n ↵+ 2 ↵+

· ·

1

1

1 , 2 1 , ↵< , ↵ 2 ↵

↵ ↵

, ↵

Proof. First note that 1/2 < b  1. Next, let f 2 An \ F (b) and consider the function 1  z f 0 (z) p(z) = ↵, 1 ↵ f (z) which is analytic in D with p(0) = 1. For this function, with ⇢ s(1 ↵) 1 b (r, s) = + (1 ↵)r + ↵ and ⌦ = ! : Re ! > , (1 ↵)r + ↵ 2 we have z f 00 (z) (p(z), zp0 (z)) = 1 + 0 2 ⌦ (z 2 D). f (z) Therefore, in view of Lemma 2.4, in order to prove that f 2 S⇤ (↵) it is enough to show that (ix, y; z) < ⌦, that is, y↵(1 ↵) 1 b +↵ Re (ix, y; z) = 2 2 2 2 (1 ↵) x + ↵ or, equivalently, ✓1 ◆✓ ↵ 1 ↵ 2◆ b ↵ y + ·x (3.1) 2 1 ↵ ↵ for all x 2 R, y  n(1 + x2 )/2 and z 2 D. This happens only when ✓1 ◆✓ ↵ n 1 ↵ 2◆ b ↵ (1 + x2 )  + ·x , 2 2 1 ↵ ↵ that is, when ✓1 ◆ ✓ ◆ b ↵ ↵ + n + 1 b ↵ 1 ↵ + n x2 0 2 1 ↵ 2 2 ↵ 2 for all x 2 R. The last inequality holds if and only if ✓1 ◆ b ↵ ↵ +n 0 2 1 ↵ 2 and ✓1 ◆ b ↵ 1 ↵ + n 0. 2 ↵ 2 b Finally, it easy to verify that satisfies the two inequalities above. ⇤

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By specifying values of ↵ and n in Theorem 3.1, we deduce the following results. Corollary 3.2. (i) (ii) (iii) (iv)

C = F (1/2) ⇢ S⇤ (since b = (0, 1) = 1/2); An \ F (b) ⇢ S⇤ (1/2) for b = min{n/2, 1}; C = F (1/2) ⇢ S⇤ (1/2) (taking n = 1 in (ii)); A2 \ F (1) ⇢ S⇤ (1/2) (taking n = 2 in (ii)).

We note that (iii) is the well-known Marx–Strohh¨acker theorem [13] and that (iv) corresponds to [8, Theorem 2.6i, page 68].

3.1. Coefficients. In [11], Ponnusamy et al. gave sharp coefficient bounds and some distortion theorems for f 2 F (1). It was also shown that every partial sum (or section) P sn (z) = z + nk=2 ak zk of a function f 2 F (1) given by (1.1) belongs to C in the disc |z| < 1/6 and that this radius is the best possible. We extend the coefficient result by finding sharp bounds for the coefficients of the Ozaki close-to-convex functions FO ( ). Theorem 3.3. Let f 2 FO ( ) be given by (1.1). Then, for n n 1 Y |an |  (k + 2 1). n! k=2 The inequality is sharp when f (z) = f (z) = (1/2 )((1/(1 Proof. Write 1+ and let

2,

z)2 )

1).

1

X z f 00 (z) = 1 + cn zn := h(z) f 0 (z) n=1

2  p(z) = h(z) 1+2

1 + 2

=1+

1 X

pn zn .

n=1

Then Re p(z) > 0 for z 2 D, Re h(z) > 1/2 and |pn |  2 for n 1 and, since cn = (1/2 + )pn , we have |cn |  1 + 2 for n 1. For each integer n, the coefficients an are polynomials with positive coefficients in cn , so |an | will be less than or equal to the result of replacing |cn | by 1 + 2 . Thus, by the principle of majorisation (see, for example, [7]), z f 00 (z) 1+2 z 1+ 0 ⌧ f (z) 1 z and 1 ◆ X 1✓ 1 f (z) ⌧ 1 := z + dn zn . 2 (1 z)2 n=2 Therefore,

n

|an |  dn = which is (3.1).

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3.2. Distortion theorems. We next give distortion results for functions f 2 FO ( ). Theorem 3.4. Let f 2 FO ( ). Then, for z = rei✓ 2 D,

Proof. From (1.3),

z f 00 (z) (1 + 2 )r  , f 0 (z) 1 r 1 1  | f 0 (z)|  , 1+2 (1 + r) (1 r)1+2 ◆ 1✓ 1 1✓ 1 1  | f (z)|  2 2 (1 + r) 2 (1 r)2 1+

Thus,

◆ z f 00 (z) ✓ 1 1 = + p(z) + f 0 (z) 2 2 1+

and so

z f 00 (z) f 0 (z)

z f 00 (z) f 0 (z)

Hence,

◆ 1. (3.2)

.

1+2 z 1 z (1 + 2 )z . 1 z

z f 00 (z) (1 + 2 )!(z) = , f 0 (z) 1 !(z)

where |!(z)|  |z|. The first inequality in the theorem now follows. To prove the inequalities for | f 0 (z)|, we use a result of Su↵ridge [14, Theorem 3], which states that if F is convex and zG0 (z) zF 0 (z), then G(z) F(z). Using this result, we integrate (3.2) to obtain 1 f 0 (z) . (1 z)1+2 The inequalities for | f 0 (z)| now follow in the same way. An application of Lemma 2.5 gives the bounds for | f (z)|.



3.3. Growth and area estimates. For f 2 S, z = rei✓ 2 D, let M(r) = max|z|=r | f (z)|, C(r) be the curve f (|z| = r), L(r) the length of C(r) and A(r) the area enclosed by C(r). A long-standing problem for functions in K is whether M(r) can be replaced by p A(r) in the growth estimate L(r) = O(M(r) log(1/(1 r))) p as r ! 1, a result already known for functions in S⇤ . Similarly, replacing M(r) by A(r) in the known estimate nan = O(M((n + 1)/n)) as n ! 1 for functions in K remains an open question [15, 16]. Since the definition of Ozaki close-to-convex functions does not include an independent starlike function, it is relatively easy to show that both these growth estimates can be improved when f 2 FO ( ), as follows.

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Theorem 3.5. Let f 2 FO ( ) be given by (1.1), with M(r), L(r) and A(r) defined as above. Then ✓p 1 ◆ L(r) = O A(r) log as r ! 1 1 r and

⇣p ⌘ nan = O A((n + 1)/n)

as n ! 1.

Proof. For z = rei✓ , L(r) =

Z

2⇡ 0

0

|z f (z)| d✓ 

Z rZ 0

2⇡ 0

|z f 00 (z) + f 0 (z)| d✓ d⇢,

where now z = ⇢ei✓ . Thus, from (3.2), ✓1

◆Z r Z

2⇡

| f 0 (z)p(z)| d✓ d⇢ + 2 0 0 ✓1 ◆ ✓ 1◆ = + I1 (r) + I2 (r), say. 2 2

L(r) 

+



1◆ 2

Z rZ 0

2⇡ 0

| f 0 (z)| d✓ d⇢

We first deal with I1 (r). The Cauchy–Schwarz inequality gives ✓Z r Z

◆1/2 ✓ Z r Z 2⇡ ◆1/2 I1 (r)  | f (z)| d✓ d⇢ |p(z)|2 d✓ d⇢ 0 0 0 0 ✓p 1 ◆ = O A(r) log as r ! 1, 1 r 2⇡

0

2

R 2⇡ p since the first integral is A(r) and since 0 |p(z)|2 d✓  2⇡(1 + 3r2 )/(1 r2 ) when p 2 P (see, for example, [10]). Applying the Cauchy–Schwarz inequality to I2 (r) gives p A(r), which therefore establishes the first estimate in Theorem 3.4. For the second estimate, we use Cauchy’s theorem to write, with z = rei✓ , 1 n an = 2⇡rn 2

Z

2⇡ 0

z(z f 0 (z))0 e

in✓

d✓

and so Z 2⇡ Z 1+2 2 1 2⇡ 0 0 n |an |  | f (z)p(z)| d✓ + | f (z)| d✓ 4⇡rn 1 0 4⇡rn 1 0 1+2 2 1 = J1 (r) + J2 (r), say. 4⇡rn 1 4⇡rn 1 2

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For J1 (r), the Cauchy–Schwarz inequality and Parseval’s theorem give J1 (r) 

✓Z

2⇡ 0

| f 0 (z)|2 d✓

◆1/2 ✓Z

2⇡

|p(z)|2 d✓

0

1 ✓ X ◆1/2 ✓ Z 2 2 2k 2 = 2⇡ k |ak | r k=1

2⇡ 0

|p(z)|2 d✓

1 ✓ X ◆1/2 ✓Z  2⇡ k|ak |2 rk (max krk 2 ) k=1

✓ A( pr) ◆1/2 ✓ 1 + 3r2 ◆1/2  2⇡ 2 , er (1 r) 1 r2 since krk

2

 1/(er2 (1

r)), again using

Finally, we note that J2 (r) =

Z

2⇡ 0

R 2⇡ 0

| f 0 (z)| d✓ 

◆1/2

2⇡ 0

◆1/2

|p(z)|2 d✓

◆1/2

|p(z)|2 d✓  2⇡(1 + 3r2 )/(1

p

2⇡

✓Z

2⇡ 0

| f 0 (z)|2 d✓

◆1/2

r2 ).

,

p which is the first expression above. Noting that A( r) = O(A(r)) as r ! 1, and choosing r = (n + 1)/n in the estimates for J1 (r) and J2 (r), the second estimate in Theorem 3.4 follows. ⇤

4. The initial coefficients of functions in FO ( , ↵) From (1.4), we can write 1+

◆ z f 00 (z) ✓ 1 1 = + p(z) + f 0 (z) 2 2

and so, by equating coefficients, a2 = (1 + 2 )p1 , 4 ✓ 1 a3 = (1 + 2 ) p2 (1 12 2 ✓ 1 a4 = (1 + 2 ) p3 (4 24 4 1 + (8 21 + 16 24

2

◆ ,

2

2

)p21

7

6

)p1 p2

18

+ 30

(4.1) 2

+ 12

2 2

◆ )p31 .

We now obtain sharp bounds for the coefficients a2 , a3 and a4 .

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Theorem 4.1. Let f 2 FO ( , ) and suppose that f is given by (1.1) for z 2 D. Then 8 1 > > > (1 + 2 ), 0<  , > > > 6 2(1 + ) < |a2 |  (1 + 2 ), |a3 |  > > 2 > 2 1 > > >   1, : (1 + )(1 + 2 ), 3 2(1 + ) r 8 > 2 > > > (1 + 2 ), 0<  , > > > 12 8 + 15 + 6 2 < |a4 |  > > r > > 2 > > 2 2 2 2 > +6 ),   1. : (1 + 2 )(1 + 8 + 15 36 8 + 15 + 6 2 All the inequalities are sharp.

Proof. The inequality for |a2 | is trivial, since |p1 |  2, and is sharp when p1 = 2. For a3 , we note that since 0  1 2 2  2 when 0 <  1/(2(1 + )), and 1 2 2 < 0 when 1/(2(1 + )) <  1, the inequalities for |a3 | follow on applying Lemma 2.1. The first inequality for a3 is sharp when p1 = 0 and p2 = 2, and the second is sharp when p1 = 2 and p2 = 2. For a4 , we will use Lemma 2.2. In the expression for a4 in (4.1), let B = (4

7

6

)/8

and

D = (8

21 + 16

2

18

p

+ 30

2

+ 12

so that 0  B  1 and B(2B 1)  D  B when 0 <  2/(8 + 15 + 6 applying Lemma 2.2 gives the first inequality for |a4 |. Next, write

2 2

)/24,

2 ).

Thus,

(1 + 2 )[p3 2Bp1 p2 + Bp31 + (D B)p31 ] p and note that D B 0 when 2/(8 + 15 + 6 2 )   4/(7 + 6 ). Thus, applying Lemma 2.2 in the case D = B gives the second bound for |a4 |, provided p 2/(8 + 15 + 6 2 )   4/(7 + 6 ). Finally, noting that the coefficients of p1 p2 and p31 in the expression for a4 in (4.1) are positive when 4/(7 + 6 )   1, and using the inequalities |pn |  2 for n = 1, 2 and 3, gives the second inequality for |a4 | in this interval. The first inequality for a4 is sharp when p1 = 0, and the second is sharp when p1 = p2 = p3 = 2. ⇤ a4 =

1 24

5. Inverse coefficients of functions in FO ( , )

For any univalent function f , there exists an inverse function f disc |!| < r0 ( f ) with Taylor expansion

1

defined on some

f 1 (!) = ! + A2 !2 + A3 !3 + A4 !4 + · · · .

(5.1)

Since FO ( , ) ⇢ S, inverse coefficients exist for functions f 2 FO ( , ). It is an easy exercise to show from (5.1) that A2 = a2 , A3 = 2a2 2 a3 , A4 = 5a2 3 + 5a2 a3

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which, on substituting from (4.1), produces A2 = A3 = A4 =

(1 + 2 )p1 , ✓ (1 + 2 ) p2 12 ✓ (1 + 2 ) p3 24 1 + (8 + 9 + 24 4

◆ 1 (1 + + 4 )p21 , 2 1 (4 + 3 + 14 )p1 p2 4 2 + 42 + 30 2 + 72

(5.2) 2 2

)p31

We can now prove the following result.

◆ .

Theorem 5.1. Let f 2 FO ( , ), with inverse function f 1 given by (5.1). Then 8 1 > > > (1 + 2 ), 0<  , > > > 1+4 > 2 > 2 1 > > >   1, : (1 + 2 )(1 + 4 ), 6 1+4 r 8 > 1 > > > (1 + 2 ), 0< 2 , > > > 1 + 30 + 72 2 < 12 |A4 |  > r > > > 1 > > 2 2 2 2 > (1 + 2 )(2 + + 30 + 72 ), 2   1. : 72 1 + 30 + 72 2 All the inequalities are sharp.

Proof. The inequality for |A2 | is obvious and is sharp when p1 = 2. For A3 , 1 |A3 |  (1 + 2 ) p2 (1 + + 4 )p21 12 2 and an application of Lemma 2.1 easily gives the inequalities for |A3 |, the first of which is sharp when p1 = 0 and p2 = 2, and the second when p1 = 2 and p2 = 2. For A4 , from (5.2),  1 A4 = (1 + 2 ) p3 (4 + 3 + 14 )p1 p2 24 4 1 + (8 + 9 + 2 + 42 + 30 2 + 72 2 2 )p31 . 24 We will use Lemma 2.2 with B = 18 (4 + 3 + 14

)

and

D=

1 24 (8

Thus, 0  B  1 when either 0
1 when 4/(3 + 14 ) <  1, and 2µ 1 0 when 2/(3 + 14 )   1 (which contains the interval 4/(3 + 14 ) <  1). So applying Lemma 2.3 gives the second inequality for |A4 | when 4/(3 + 14 ) <  1. The first inequality for A4 is sharp when p1 = 0, and the second is sharp when p1 = 2, p2 = 2 and p3 = 2. ⇤

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VASUDEVARAO ALLU, NFA-18, IIT Campus, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India e-mail: [email protected] DEREK K. THOMAS, Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK e-mail: [email protected] NIKOLA TUNESKI, Faculty of Mechanical Engineering, Ss. Cyril and Methodius University in Skopje, Karpos 2 bb, 1000 Skopje, Republic of Macedonia e-mail: [email protected]

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