On neighborhoods of functions associated with conic domains

(1)

On neighborhoods of functions associated with conic domains

Nihat Ya˘gmur

Abstract

Letk−ST[A, B], k≥0,−1≤B < A≤1 be the class of normalized analytic functions defined in the open unit disk satisfying

<





(B−1)^zf_f⁰_(z)^(z) −(A−1) (B+ 1)^zf_f⁰_(z)^(z) −(A+ 1)



> k

(B−1)^zf_f_(z)⁰^(z)−(A−1) (B+ 1)^zf_f_(z)⁰^(z)−(A+ 1)−1

.

and letk−U CV[A, B], k≥0,−1≤B < A≤1 be the corresponding class satisfying

<







(B−1)(^zf⁰^(z))⁰

f⁰(z) −(A−1) (B+ 1)^(zf_f0⁰(z)^(z))⁰ −(A+ 1)





> k

(B−1)(^zf⁰^(z))⁰

f⁰(z) −(A−1) (B+ 1)^(zf_f0⁰(z)^(z))⁰ −(A+ 1) −1

.

For an appropriate δ > 0, the δ neighborhood of a function f ∈ k− U CV[A, B] is shown to consist of functions in the classk−ST[A, B].

1 Introduction

LetAdenote the family of functionsf of the form f(z) =z+

∞

X

n=2

anzⁿ (1)

Key Words: Analytic function,k−starlike functions,k−uniformly convex functions.

2010 Mathematics Subject Classification: 30C45.

Received: 8 May, 2014.

Accepted: 30 June, 2014.

291

(2)

which are analytic in the open unit disk U = {z:|z|<1}. The classes S^∗ and C are the well-known classes of starlike and convex univalent functions respectively; for details, see[4].

Noor and Malik introduced and studied the classk−U CV[A, B] and the corresponding classk−ST[A, B] in [10] as following:

A function f(z)∈Ais said to be in the classk−ST[A, B], k ≥0,−1≤ B < A≤1,if and only if,

<





(B−1)^zf_f(z)⁰^(z)−(A−1) (B+ 1)^zf_f(z)⁰^(z)−(A+ 1)



> k

(B−1)^zf_f(z)⁰^(z)−(A−1) (B+ 1)^zf_f(z)⁰^(z)−(A+ 1)−1

. (2)

A function f(z)∈ Ais said to be in the classk−U CV[A, B], k≥0,−1 ≤ B < A≤1,if and only if,

<







(B−1)(^zf⁰^(z))⁰

f⁰(z) −(A−1) (B+ 1)^(zf_f0⁰(z)^(z))⁰ −(A+ 1)





> k

(B−1)(^zf⁰^(z))⁰

f⁰(z) −(A−1) (B+ 1)^(zf_f0⁰(z)^(z))⁰ −(A+ 1) −1

. (3)

It can be easily seen that

f(z)∈k−U CV[A, B]⇐⇒zf⁰(z)∈k−ST[A, B]. (4) Special cases.

i. k−ST[1,−1] =k−ST, k−U CV[1,−1] =k−U CV,the well-known classes of k−starlike and k−uniformly convex functions respectively, introduced by Kanas and Wisniowska [6, 7].

ii. k−ST[1−2α,−1] =SD(k, α), k−U CV[1−2α,−1] =KD(k, α),the classes introduced by Shams et al. in [17].

iii. 0−ST[A, B] = S^∗[A, B], 0−U CV[A, B] = C[A, B], the well-known classes of Janowski starlike and Janowski convex functions respectively, introduced by Janowski [5].

Geometrically, if a functionf(z)∈k−ST[A, B] then ^(B−1)

zf0(z) f(z) −(A−1) (B+1)^zf_f(z)⁰^(z)−(A+1) = wtakes all values from the domain Ω_k, k≥0, as

Ω_k={w:<w > k|w−1|} (5) or equivalently

Ωk=n

u+iv:u > kp

(u−1)²+v²o

. (6)

(3)

The domain Ωk represents the right half plane for k= 0, a hyperbola for 0< k <1,a parabola fork= 1 and an ellipse fork >1, for more details see [10].

Given δ ≥ 0, Ruscheweyh [16] defined the δ−neighborhood Nδ(f) of a functionf(z)∈Aby

Nδ(f) = (

g(z) :g(z) =z+

∞

X

n=2

bnzⁿ,and

∞

X

n=2

n|an−bn| ≤δ )

. (7) Ruscheweyh [16] proved among other results thatN_1/4(f)⊂S^∗forf ∈C.

Sheil-Small and Silvia [18] introduced more general notions of neighborhood of an analytic function. These included noncoefficient neighbourhoods as well.

Problems related to the neighborhoods and some other properties of analytic functions were considered by many authors, for example, see [1, 2, 3, 8, 9, 11, 12, 13, 14].

In this paper, the neighborhoodNδ(f) for functionsf(z)∈k−U CV[A, B]

is investigated. It is shown that all functions g ∈ Nδ(f) are in the class k−ST[A, B] for certainδ >0.

2 Main Results

In order to obtain the main results, a characterization of the classk−ST[A, B]

in terms of the functions in another classk−ST⁰[A, B] is needed. Forz∈U, a functionHtis said to be in the classk−ST⁰[A, B] if the functionHtis of the form

Ht(z) = 1

(B−A)

1−kt± q

t²−(kt−1)²i

×





 z

B−1−(B+ 1)

kt± q

t²−(kt−1)²i

(1−z)²

− z

A−1−(A+ 1) (kt± q

t²−(kt−1)²i)

1−z







(8)

where

t≥_k+1¹ , if 0≤k≤1,

1

k+1 ≤ t≤_k−1¹ , if k >1. (9)

(4)

Recall that for any two functionsf(z) andg(z) given by f(z) =z+

∞

X

n=2

a_nzⁿ, g(z) =z+

∞

X

n=2

b_nzⁿ

the Hadamard product (or convolution) off andg is defined by (f∗g)(z) =z+

∞

X

n=2

anbnzⁿ=(g∗f)(z) (10) Lemma 2.1. A functionf is in the classk−ST[A, B]if and only if

1

z(f∗Ht)(z)6= 0, z∈U (11) for allHt∈k−ST⁰[A, B].

Proof. Let f ∈ k − ST[A, B]. Then the image of U under

w = u + iv = ^(B−1)

zf0(z) f(z) −(A−1)

(B+1)^zf_f(z)⁰^(z)−(A+1) lies in the conic type regions Ωk =n

u+iv:u > kp

(u−1)²+v²o

so that forz∈U (B−1)^zf_f(z)⁰^(z)−(A−1)

(B+ 1)^zf_f(z)⁰^(z)−(A+ 1) 6=kt± q

t²−(kt−1)²i (12) where

t≥_k+1¹ , if 0≤k≤1,

1

k+1 ≤ t≤_k−1¹ , if k >1. (13) Thusf ∈k−ST[A, B] if and only if

B−1−(B+ 1) (kt± q

t²−(kt−1)²i)

zf⁰(z)

−

A−1−(A+ 1) (kt± q

t²−(kt−1)²i)

f(z)

/

z(B−A)

1−kt± q

t²−(kt−1)²i

6= 0 (14) or equivalently

1

z(f ∗Ht)(z)6= 0, z∈U for allHt∈k−ST⁰[A, B].

(5)

Lemma 2.2. If

Ht(z) =z+

∞

X

n=2

hn(t)zⁿ ∈k−ST⁰[A, B], z∈U (15) then

|hn(t)|< 2n(t+ 1)

|B−A|t (16)

wheret is given in (13).

Proof. Writing Ht(z) =z+

∞

P

n=2

hn(t)zⁿ,and comparing coefficients ofzⁿ in (8) we obtain

h_n(t) = [n(B−1)−(A−1)−(n(B+ 1)−(A+ 1))kt

±(n(B+ 1)−(A+ 1)) q

t²−(kt−1)²i]

/

(B−A)

1−kt± q

t²−(kt−1)²i

(17) Thus

|hn(t)|² = [|n(B−1)−(A−1)−(n(B+ 1)−(A+ 1))kt

±(n(B+ 1)−(A+ 1)) q

t²−(kt−1)²i ²] /

(B−A)

1−kt± q

t²−(kt−1)²i

2

= h

{n(B−1)−(A−1)−[n(B+ 1)−(A+ 1)]kt}² +(n(B+ 1)−(A+ 1))²

t²−(kt−1)²i /h

(B−A)²

(1−kt)²+t²−(kt−1)²i

= 4(n−1) [(nB−A) (kt−1) + (n−1)kt] + [nB−A+n−1]²t²

(B−A)²t² .

If−1≤B < A≤1, n≥2 thennB−A < n−1,and from (13),kt−1< t.

So

|hn(t)|²≤ 4(n−1)²(t+ 1)²

(B−A)²t² < 4n²(t+ 1)²

(B−A)²t² (18)

and

|hn(t)|<2n(t+ 1)

|B−A|t.

(6)

Lemma 2.3. For a functionf ∈Aandε∈C,define the functionFε by

F_ε=f(z) +εz

1 +ε . (19)

If for every ε, |ε| < δ we have Fε ∈ k−ST[A, B], then for every Ht ∈ k−ST⁰[A, B]

1

z(f∗H_t)(z)

≥δ, z∈U. (20) Proof. If F_ε ∈ k−ST[A, B] for every ε, |ε| < δ, then by Lemma 2.1 for all H_t∈k−ST⁰[A, B]

1

z(F_ε∗H_t)(z)6= 0, z∈U or equivalently

(f∗Ht)(z) +εz

(1 +ε)z 6= 0. (21)

Since|ε|< δ,it is easily follows that 1

z(f ∗H_t)(z)

≥δ.

Theorem 2.1. Let f ∈A,ε∈C andδ >0. If for every ε,|ε|< δ,we have Fε∈k−ST[A, B] then

Nδ⁰(f)⊂k−ST[A, B] (22)

for

δ⁰ =|B−A|t

2(t+ 1) δ (23)

where−1≤B < A≤1 and

t≥_k+1¹ , if 0≤k≤1,

1

k+1 ≤ t≤_k−1¹ , if k >1.

Proof. Letg(z) =z+

∞

P

n=2

bnzⁿ∈Nδ⁰(f).For anyHt∈k−ST⁰[A, B],

1

z(g∗H_t)(z)

= 1

z(f ∗H_t)(z) +1

z((g−f)∗H_t) (z)

≥ 1

z(f ∗Ht)(z)

− 1

z((g−f)∗Ht) (z)

(7)

From lemma 2.3, 1

z(g∗H_t)(z)

≥ δ−

∞

X

n=2

(b_n−a_n)h_n(t)zⁿ z

≥ δ−

∞

X

n=2

|(bn−a_n)| |hn(t)|

≥ δ− 2(t+ 1)

|B−A|t

∞

X

n=2

n|(bn−a_n)|.

Using Lemma 2.2 and noting thatg∈Nδ⁰(f) and whence

∞

P

n=2

n|(bn−a_n)|< δ⁰,

thus

1

z(g∗Ht)(z)

≥δ− 2(t+ 1)

|B−A|tδ⁰= 0 Therefore

¹_z(g∗Ht)(z)

6= 0 inUfor allHt∈k−ST⁰[A, B] if δ⁰ =|B−A|t

2(t+ 1) δ

By Lemma 2.1g∈k−ST[A, B].This proves thatNδ⁰(f)⊂k−ST[A, B].

Lemma 2.4. ([15])Ifφis a convex univalent function withφ(0) = 0 =φ⁰(0)−

1 in Uand g is starlike univalent in U, then for each analytic function F in U,

(φ∗F g)(z)

(φ∗g)(z) ⊂coF(U), (z∈U) (24) wherecostands for the closed convex hull.

Lemma 2.5. If φ(z)∈C, f(z)∈k−ST[A, B] andg(z) = (B+ 1)zf⁰(z)− (A+ 1)f(z)∈S^∗ then(φ∗f)(z)∈k−ST[A, B].

Proof. AssumeF(z) = ^(B−1)

zf0(z) f(z) −(A−1)

(B+1)^zf_f(z)⁰^(z)−(A+1).Then forG(z) = (φ∗f)(z), we have zG⁰(z) =φ(z)∗zf⁰(z).

(8)

Hence

(B−1)^zG_G(z)⁰^(z)−(A−1) (B+ 1)^zG_G(z)⁰^(z)−(A+ 1) =

(B−1)^φ(z)∗zf_(φ∗f)(z)⁰^(z)−(A−1) (B+ 1)^φ(z)∗zf_(φ∗f)(z)⁰^(z)−(A+ 1)

= φ(z)∗[(B−1)zf⁰(z)−(A−1)f(z)]

φ(z)∗[(B+ 1)zf⁰(z)−(A+ 1)f(z)]

=

φ(z)∗

(B−1)^zf_f(z)⁰^(z)−(A−1) (B+1)^zf_f(z)⁰^(z)−(A+1)

[(B+ 1)zf⁰(z)−(A+ 1)f(z)]

φ(z)∗[(B+ 1)zf⁰(z)−(A+ 1)f(z)]

= φ(z)∗(F g)(z)

φ(z)∗g(z) . (25)

By Lemma 2.4 the image ofUunder φ(z)∗(F g)(z)

φ(z)∗g(z) is a subset of the convex hull ofF(U).That means (φ∗f)(z)∈k−ST[A, B].

Theorem 2.2. Iff ∈k−U CV[A, B]andg(z) = (B+1)zf⁰(z)−(A+1)f(z)∈ C thenFε= ^f(z)+εz_1+ε ∈k−ST[A, B]for|ε|< ¹₄.

Proof. Letf(z) =z+

∞

P

n=2

a_nzⁿ∈k−U CV[A, B].Then Fε = f(z) +εz

1 +ε

=

z(1 +ε) +

∞

P

n=2

a_nzⁿ 1 +ε

=

f(z)∗

z(1 +ε) +

∞

P

n=2

zⁿ

1 +ε

= f(z)∗

z−_1+ε^ε z²

1−z =f(z)∗h(z) whereh(z) =^z−

ε 1+εz² 1−z .Now, zh⁰(z)

h(z) = 1−2ρz+ρz²

(1−ρz) (1−z), where ρ= ε 1 +ε. Hence|ρ|= _1−|ε|^|ε| <¹₃ gives|ε|< ¹₄. Thus

<

zh⁰(z) h(z)

≥ 1−2|ρ| |z| − |ρ| |z|²

(1− |ρ| |z|) (1 +|z|) >0. (26)

(9)

This inequality holds for all |ρ|< ¹₃ which is true for |ε|< ¹₄. Thereforehis starlike inUand so

z

Z

0

h(t)

t dt=z+

∞

X

n=2

c_n

nzⁿ=h(z)∗log 1

1−z

(27)

is convex for|ε|< ¹₄

(f∗h) (z) =zf⁰(z)∗

h(z)∗log 1

1−z

(28)

f(z)∈k−U CV[A, B] =⇒zf⁰(z)∈k−ST[A, B] and h(z)∗log 1

1−z

∈C.

Alsog(z)∈C implies thatzg⁰(z)∈S^∗.By Lemma 2.5, we have zf⁰(z)∗

h(z)∗log 1

1−z

∈k−ST[A, B].

Thus

(f∗h) (z) =f(z) +εz

1 +ε ∈k−ST[A, B].

Theorem 2.3. Let f ∈k−U CV[A, B] then

Nδ⁰(f)⊂k−ST[A, B] (29)

for

δ⁰= |B−A|t

8(t+ 1) (30)

where−1≤B < A≤1 and

t≥_k+1¹ , if 0≤k≤1,

1

k+1 ≤ t≤_k−1¹ , if k >1.

Proof. Let f ∈ k−U CV[A, B] then we get the result by taking δ = ¹₄ in Theorem 2.1 and using Theorem 2.2.

References

[1] R.M. Ali, K.G. Subramanian, V. Ravichandran and O.P. Ahuja, Neigh- borhoods of starlike and convex functions associated with parabola,J. Ineq.

and Appl., 10.1155(2008) 346279.

(10)

[2] O. Altınta¸s and S. Owa,Neighborhoods of certain analytic functions with negative coefficients,Int. J. Math. Sci.,19(1996) 797-800.

[3] O. Altınta¸s, H. Irmak, H. M. Srivastava, Neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator,Comput. Math. Appl.55(2008), no. 3, 331–338.

[4] A.W. Goodman,Univalent Functions,vols. I-II, Mariner Publishing Com- pany, Tempa, Florida, USA, 1983.

[5] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math.28(1973) 297-326.

[6] S. Kanas and A. Wisniowska, Conic regions and k−uniform convexity, J. Comput. Appl. Math.,105(1999) 327-336.

[7] S. Kanas and A. Wisniowska,Conic domains and starlike functions,Rev.

Roumaine Math. Pures Appl., 45(2000) 647-657.

[8] B. S. Keerthi, A. Gangadharan, H. M. Srivastava,Neighborhoods of certain subclasses of analytic functions of complex order with negative coefficients,Math. Comput. Modelling 47(2008), no. 3-4, 271-277.

[9] G. Murugusundaramoorthy, H. M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure Appl.

Math. 5(2004), no. 2, Article 24, 8 pp. (electronic).

[10] K.I. Noor, S.N. Malik,On coefficient inequalities of functions associated with conic domains, Comput. Math. Appl.62, (2011), no.5, 2209-2217.

[11] H. Orhan, E. Kadio˘glu, Neighborhoods of a class of analytic functions with negative coefficients, Tamsui Oxf. J. Math. Sci. 20 (2004), no. 2, 135-142.

[12] H. Orhan, M. Kamali, Neighborhoods of a class of analytic functions with negative coefficients, Acta Math. Acad. Paedagog. Nyh´azi (N.S.) 21 (2005), no. 1, 55–61 (electronic).

[13] H. Orhan, On neighborhoods of analytic functions defined by using Hadamard product,Novi Sad J. Math.37 (2007), no. 1, 17-25.

[14] H. Orhan,Neighborhoods of a certain class of p-valent functions with negative coefficients defined by using a differential operator, Math. Inequal.

Appl.12 (2009), no. 2, 335-349.

(11)

[15] St. Ruscheweyh and T. Sheil-Small,Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture,Comm. Math. Helv.,48(1973) 119-135.

[16] St. Ruscheweyh,Neighborhoods of univalent functions,Proceedings of the American Mathematical Society,81(1981) 521-527.

[17] S. Shams, S.R. Kulkarni, J.M. Jahangiri, Classes of uniformly starlike and convex functions, Int. J. Math. & Math. Sci.55 (2004) 2959-2961.

[18] T. Sheil-Small and E.M. Silvia,Neighborhoods of analytic functions,Jour- nal d’Analyse Mathematique, vol.52, (1989) 210-240.

Nihat Ya˘gmur,

Department of Mathematics, Faculty of Science and Art, Erzincan University, 24000, Erzincan, Turkey.

Email: [email protected]

(12)