Certain Properties of Parabolic Starlike and Convex Functions of Order

MALAYSIAN MATHEMATICAL

SCIENCES SOCIETY

Certain Properties of Parabolic Starlike and Convex Functions of Order ρ

R.AGHALARY AND S.R.KULKARNI

Department of Mathematics, Fergusson College, Pune, Maharashtra, India 411 004 e-mail: [email protected]

Abstract. We investigate starlike and convex functions ⁿ n anz z

z

f = +∑^∞₌

) 2

( with the

property that ) (

) (

z f

z f z ′

and ) (

) 1 (

z f

z f z

′

+ ′ lie inside a certain parabola. We give some particular

examples of functions having the required properties and it is shown that these functions are invariant under particular integral operators. We also determine the radii of uniformally convexity and starlikeness for certain functions. Such type of work was carried out by [1] and we are motivated by this work.

1. Introduction

Let

A

denote the class of all functions f(z) which are analytic in the open unit disc

{

^{: <1}

}

= z z

U and f(0)= f′(0)−1=0 and let S denote the subclass of A consisting of functions which are also univalent in U. A great deal of attention has been given in recent years to the uniformally starlike and convex functions introduced by Goodman [4].

He introduced the class UCV of uniformally convex functions which have the additional property that for every circular arc γ contained in U with center also in U the image arc

) (γ

f is convex.

Ma and Minda [7] and Ronning [9] independently proved that f ∈UCV if and only if

. ) ) (

( ) 1 (

) (

)

Re ( z U

z f

z f z z

f z f

z ∈

′

> ′′

′ +

′′ (1.1)

Furthermore Ronning [9] defined the class S_p of functions f ∈A for which .

) ( ) 1

( ) ( )

( )

Re ( z U

z f

z f z z f

z f

z − ∈

′

> ′

′ (1.2)

In [1] the class S_p was generalized by introducing a parameter ρ. For 0≤ ρ <1, let Ω_p be the region

{

u+iv:v² ≤ 4(1− )(u− )

}

=

{

w: w−1 ≤1−2 +Rew

}

=

Ω_ρ ρ ρ ρ (1.3)

and suppose S_p(ρ) be the subclass of A consisting of functions f such that )

) ( (

)

( z U

z f

z f

z ′ ∈Ω ∈

ρ

and also let K_p(ρ)be the subclass of A consisting of functions f such that zf′∈S_p(ρ).

It is easily seen that S_p(1/2) = S_p and S_p(ρ) is a subset of starlike functions.

A function f belonging to S_p(ρ) and K_p(ρ) is called parabolic starlike and convex of order ρ,respectively.

In [1] Ali and Singh, obtained sharp upper bounds for n-th coefficient of functions in (ρ)

Sp and for the inverse function f⁻¹(w) = w+d₂w² + when n = 2,3,and 4.

Also they obtained a general Littlewood type of bound on a_n .

Motivated by the work of Ali and Singh [1] we study the radius problem for certain functions and we introduce some examples for the classes S_p(ρ)and K_p(ρ).

It is also shown that these classes are invariant under particular integral operators.

Further results obtained in [10] will be special cases of our results.

2. Integral operators

The function which maps U onto the parabolic region Ω_ρ is given by

1 . log1 ) 1 ( 1 4 ) (

2

2 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + + −

=

z z z

q π

ρ ρ

(The branch of z is chosen such that Im z ≥ 0.)

It is clear then that f ∈S_p(ρ) and K_p(ρ), respectively, if and only if, ^z_f^f₍^′(_z^z₎⁾ and

)

1+ _f^z_′^f(_z^′′ are subordinate to q_ρ(z), (see [1]), which we denote by

. ) ) (

( ) 1 (

, ) ) (

( )

( q z

z f

z f z z

z q f

z f

z ≺ ρ ≺ ρ

′ + ′′

′

The convolution of two power series

∑

^∞

=

=

0

) (

n n nz a z

f and

∑

^∞

=

=

0

) (

n n nz b z

g

is defined as the power series

, )

)(

* (

0

∑

^∞

=

=

n

n n nb z a z

g f

In our next investigation we need the following Lemma of Ruscheweyh [12].

Lemma 1 ([12]). Let ϕ be convex and g be starlike. Then for each function F, analytic in U, the image of U under ^ϕ_ϕ^*_*^Fg_g is a subset of the convex hull of F(U).

Theorem 1. If f ∈S_p(ρ)(orK_p(ρ)) then so is f *ϕ for any function ϕ(z)= z+ analytic and convex in U.

Proof. We know f ∈S_p(ρ) if and only if, for z∈U, . ) ) (

( )

( q z

z f

z f

z ′ ≺ ρ

But q_ρ(z) is convex and f is starlike of order .ρ So an application of Lemma 1 yields

. )

* ( )

* (

* ) (

*

z f q

f z f

f f

f z

ϕ ρ

ϕ ϕ

ϕ ′ ≺

=

′

Hence it follows that ϕ* f ∈S_p(ρ). The result for K_p(ρ) now follows from the relationship f ∈K_p(ρ) if and only if zf′∈S_p(ρ).

Corollary 1. If f ∈S_p(ρ),K_p(ρ), then so is

∫

^>

+ _{z r−}

r t f t dt r

z r

0

1 () , Re( ) 0.

1

Proof. Since

1 .

* )

1 (

0 1

1 n

z

n r

r z

r n f r dt t f t z

r

∫ ∑

^∞

=

−

+

= + +

The result follows from Theorem 1 and noting that

∑

_n^∞_{=1 n}¹⁺₊^r_r zⁿis convex in U.

See [11].

Corollary 2. If f ∈S_p(ρ),K_p(ρ), then so is

. 1 , 1 ) ,

( ) (

0 ≤ ≠

−

∫

^{z f ξ}_ξ − _x^f_ξ^{xξ dξ x x}

Proof. We may write

h f x d

x f f

z ( ) ( ) *

0 =

−

∫

^ξ_ξ − _ξ ^{ξ ξ}

where

. 1 , 1 1 ,

log 1 1

1 )

1 ( ) 1 (

1

≠

⎥⎦ ≤

⎢⎣ ⎤

⎡

−

−

= −

−

=

∑

^∞ −

=

x z x

xz z x

n x z x

h

n

n n

Since h is convex the result follows from Theorem 1.

Corollary 3. Let μ ≥ 0 and

⎪⎩

⎪⎨

⎧

≤

≤

<

− ≤

−

≤

6 / 5 2

/ 1

2 / 1 ) 0

1 ( 2

2 1

0 ρ

ρ ρ ρ ρ

if A if

Then the function

n n

n

z n

z A

f

∑

^∞

=

−

= +

1 1

) 1 ( )

( _μ

μ

belongs to K_p(ρ), 0≤ ρ ≤ ₆⁵, where ρ₀ is the smallest positive root of the equation .

0 ) 1 ( ) 2 1 ( 2 ) 2 1 ( 2 ) 1

( + ⁴ − + ³ + − + − =

− ρ x ρ x ρ x ρ

Proof. Since the function f(z)= ₁₋^z_Az belongs to K_p(ρ) with condition mentioned above for A and function θ_{μ μ}

) 1 0 (

)

( n

z n

z =

∑

^∞_{= +}n is convex (see [6]). Hence the result follows from Theorem 1.

Theorem 2. Let f_i ∈K_p(ρ) and let α_i be real numbers such that α_i ≥ 0 and

1 ≤1

∑_iⁿ= α_i . Let β ≥1 then the function g(z) defined by

ξ ξ

α βd

f z

g ^z

n i

i ⁱ

/ 1

0 1

) ( )

(

∫ ∏

⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ ′

=

=

also belongs to K_p(ρ).

Proof. Since f_i ∈K_p(ρ) we have

) (

) ( )

( ) ) (

2 2 ) (

( ) Re (

1 g z

z g z z

f z f z z

g z g

z ⁿ

i i

i

i ′

≥ ′′

′′

≥ ′′

−

⎟⎟+

⎠

⎜⎜ ⎞

⎝

⎛

′

′′

∑

=

β α

ρ β

β ,

which implies that g∈K_p(ρ).

Remark 1. The special case of Theorem 2 for ρ =1/2, β =1 was proved by Shanmugam and Ravichandran [10].

Theorem 3. Suppose f ∈A is such that ^z_f^f₍^′(_z^z₎⁾ −1 < ¹⁻_t^ρ, where t > 0 and .

1

0≤ ρ < Then

ξ ξ

ξ d

z f g

t

∫

z^⎜⎜_⎝^{⎛ ⎟⎟}_⎠^⎞

= 0

) ) (

(

belongs to K_p(ρ).

Proof. Since

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ′ −

′ =

′′ 1

) (

) ( )

( ) (

z f

z f t z z g

z g z

we can write

) . (

) 1 (

) 2 2 ) ( (

) Re (

z g

z g z z

g z g z

′

≥ ′′

−

≥

−

′ +

′′ ρ ρ

Hence g∈K_p(ρ).

Remark 2. The special case of Theorem 3 for ρ = ₂¹,t = 2 was proved by Shanmugan and Ravichandran [10].

We will need the following lemma in the next theorem.

Lemma 2 ([2]). Let β,γ ∈C. Let h(z)= c+h₁z+ be a convex (univalent) function in U, with Re(βh(z)+γ)> 0, z∈U. If p(z) = c+ p₁z+ is analytic in U, then

) ) (

( ) ) (

( h z

z p

z p z z

p ≺

γ

β +

+ ′ implies p(z) ≺h(z).

Theorem 4. Let f ∈A and α > 0, f ∈S_p(ρ) then

α α

α

/ 1 0

) ) (

( ⎥

⎦

⎢ ⎤

⎣

= ⎡

∫

^{z f} _t ^{t dt}

z

F (2.1) also belongs to S_p(ρ).

Proof. Differentiation of (2.1) w.r.t. to z, leads to

. ) 1 (

) ( )

( ) ( )

( ) ) ( 1

( ′ −

′ = + ′′

− ′

z f

z f z z

F z F z z F

z F

z α

α (2.2)

Let p(z) = ^z_F^F₍^′(_z^z₎⁾, then (2.2) is equivalent to

) . (

) ( ) (

) ( ) 1

( f z

z f z z p

z p z z

p ′

′ =

+ α (2.3)

Since f ∈S_p(ρ), it follows by (2.3)

. ) ) (

( ) ( ) 1

( ρ

α p z ^qp z p z z

p ′ ≺

+

But it is easy to see that Re(αq_p(ρ)) > 0 and q_p(ρ) is convex (univalent). By Lemma 2 it follows

. ) ( )

(z q_p ρ

p ≺

Hence F ∈S_p(ρ).

Finally in this section we investigate sections of elements K_p(ρ).

Theorem 5. If f(z)= z+

∑

_k^∞₌₂a_kz^k ∈K_p(ρ), then f_n(z) = z+ ∑ⁿ_k=2a_kz^k belongs to K_p(ρ) for z < ¹₄.

Proof. Let g_n(z) = z+

∑

ⁿ_k=2 z^k = ^z₁⁻₋^zn_z⁺¹, it is well known g_n is convex for

4.

< 1

z Hence function h_n(z) = 4g_n(z/4) is convex in U. By making use of Theorem 1 it follows that h_n ∗ f ∈K_p(ρ) or f_n ∈K_p(ρ) for z < ₄¹. Hence the proof is complete.

3. Radius problem

Let M(β) denote the class of all analytic functions p(z) defined on U, with p(0)=1 satisfying arg p(z) < ^πβ₂ , (z∈U), where β > 0.

Definition 1. A function f(z) in the class A is said to be a member of the class )

,

*(M_β α

C if and only if there is a function g(z)∈S^*(α) (starlike of order α) such that _g^f₍⁽_z^z₎⁾ ∈M(β).

Definition 2. TheK_p(ρ) radius of S denoted R_p(ρ) is the radius of the largest disc )

(ρ Rp

z < in which 1+ ^z_f^f_′^′′₍⁽_z^z₎⁾ ∈Ω_ρ holds for all f ∈S.

The S_p(ρ) radius of C*(M_β,α), denoted R^∗_p(ρ) is the radius of the largest disc )

*(ρ Rp

z < in which ^z_f^f′₍⁽_z^z₎⁾ ∈Ω_ρ holds for all f ∈C^*(M_β,α).

Let 0 ≤ ρ <1,a > ρ and let m_a(ρ) be the largest number in which disc )}

(

; { )) ( ,

(a m_a ρ w C w a m_a ρ

D = ∈ − < lies completely inside region Ω_ρ. A direct calculation gives us

. 2

if ) 1 ( ) 1 ( 2

2 if

)

( ⎪⎩

⎪⎨

⎧

−

≥

−

−

−

<

<

= −

ρ ρ

ρ ρ

ρ ρ

a a

a a

m_a

If we restrict the value of a by ¹⁺₂^ρ < a< (2− ρ)+ ρ² − ρ + ₂⁵ then disc contains the point 1. Hence it follows:

Lemma 3. Let f ∈A. If for any a with ¹⁺₂^ρ < a <5−4ρ

) ) (

( )

( a m_a ρ

z f

z f

z ′ − <

for all z∈U. Then f ∈S_p(ρ).

To prove the next theorems we need the following results.

Lemma 4. Suppose p(z) =1+c_nzⁿ + is analytic and belongs to M(β), then

n n

r nr z

p z p z

1 2

2 ) (

) (

≤ −

′ β

for z = r <1.

Proof. Define g(z) = p(z)^1/β. Then argg(z) = ¹_βargp(z) and Reg(z) > 0. It is known [8] that

, 1

2 ) (

) (

2n n

r nr z

g z g z

≤ −

′ for z = r <1.

Hence

, 1

2 ) (

) ( )

( ) (

2n n

r nr z

g z g z z

p z p z

≤ −

= ′

′ β β for z = r <1,

and the proof is complete.

Lemma 5 ([3]). If p(z)=1+c₁z +c₂z+ is analytic and Rep(z) >α for U

z∈ then

2 2

2

1 ) 1 ( 2 1

) 2 1 ( ) 1 (

r r r

z r

p −

≤ −

−

−

− + α α

for z = r <1.

Theorem 6. The K_p(ρ) radius of S is

1 . 3 ) 2

(

2

ρ ρ ρ

+ +

= − Rp

Equality occurs for ( ) ₂.

) 1 ( z

z z

f = −

Proof. For f ∈S, it is known [5] that

, 1

4 1

1 ) (

) 1 (

2 2

2

r r r

r z

f z f z

< −

−

− +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

′

+ ′′ for z = r <1.

This represents a circular disc intersecting the real axis in

2 2

1 1

4 1

r r x r

− +

= − and .

1 4 1

2 2

2 r

r x r

− +

= +

For r = R_p(ρ) we have = ρ

−+

−

2 2

1 4 1

r r

r and for r less than this value the disc lies completely inside the parabolic region Ω_ρ , by Lemma 3. Hence the proof is complete.

Theorem 7. The S_p(ρ) radius R^*_p(ρ) for the class C^∗(M_β,α) is

) 2 1

)(

1 ( ) 1

) 1

( ) 1 (

2

*

α ρ ρ α

β α

β ρ ρ

− +

−

−

− + +

− +

= − Rp

where β ≥ ₄³(1− ρ) and 0 ≤α <1,0 ≤ ρ <1.

Proof. Let g be a starlike function of order α such that h(z) = _g^f₍⁽_z^z₎⁾ ∈M(β). Since

) ,

( ) ( ) (

) ( ) (

) (

z h

z h z z g

z g z z f

z f

z′ = ′ + ′ by Lemmas 4 and 5, we get

2 2

2

1 ) 1

( 2 1

) 2 1 ( 1 ) (

) (

r r r

r z

f z f z

−

−

≤ +

−

−

− +

′ α β α

(3.1)

This circular disc touches the parabola Ω_ρ if 2 .

2

1

) 2 1 ( ) 1 ( 2

1^{− +β−α}₋ ^{+ − α} = ρ

r r r

This gives the value of R^∗_p(ρ). For r less than this value the circular disc is inside the parabola Ω_ρ. The result is sharp for function ^{( )} _{(1 )}2(1 )

( )

₁^{1 ,}

β

α z

z z

z z

f ⁺₋

− ⁻

= which satisfies

the hypothesis with ( ) _{2(1 )}.

) 1 ( − ^−α

= z

z z

g

Remark 3. The special case of Theorem 7 for ρ = ¹₂ and β =1 was proved by Shanmugam and Ravichandran [10].

Acknowledgement. The authors would like to thank the referee for his valuable suggestions for improvement of the manuscript.

References

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2. P.J. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot-Bouquet differential subordination, proceeding of the conference on General Inequalities-IV, Birkhauser Verlag, Basel. ISNM 64 (1983), 339−348.

3. A.W. Goodman, Univalent Functions, Vol. 2, Polygonal Publ. House, Washington, New Jersy, 1983.

4. A.W. Goodman, On uniformly convex functions, Ann. Polon, Math. 56 (1991), 87−92.

5. G.M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical monographs (American Mathematical Society, Providence, Rhode Island, 1969).

6. J.L. Lewis, Convexity of a certain series, J. London Math. Soc. (2) 27 (1983), 435−446.

7. W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165−175.

8. T.H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math.

Soc. 14 (1963), 521−524.

9. F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189−196.

10. T.N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions.

Seminar (CMFT, 94) copyright, 1995 by World Scientific Publishing (Editors: R.M. Ali, S.T. Ruscheweyh and E.B. Saff), 319−324.

11. S.T. Ruscheweyh, New Criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109−115.

12. S.T. Ruscheweyh and T-Shell. Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment, Math. Helv. 48 (1973), 119−135.

Keywords and phrases: starlike and convex function, Hadamard product, subordination.

1991 Mathematics Subject Classification: 30C45