ON THE GENERALIZED VECTOR EQUILIBRIUM PROBLEM

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LINEAR COMBINATIONS OF q-STARLIKE FUNCTIONS OF ORDER ALPHA

HAMID SHAMSAN1_{, READ S.A. QAHTAN}1 _{AND S. LATHA}1_,_§

Abstract. In this paper, we introduced a new concept of bounded radius rotation to define the class ofq-starlike functions of orderα using theq-derivative, some geometric properties of linear combination of such functions are studied.

Keywords:q-derivative,q-starlike functions, convex functions, linear combination, bounded radius rotation.

AMS Subject Classification: 05C40, 05C99.

1. Introduction

LetA denote the class of functions of form

f(z) =z+

∞ X

n=2

anzn, (1)

which are analytic in the open unit disk

U ={z:z∈C and|z|<1},

and S denote the subclass ofA consisting of all function which are univalent inU.

Jackson[5] initiatedq-calculus and developed the concept of the q-integral andq-derivative. For a functionf ∈ S given by (1) and 0< q <1, theq-derivative off is defined by

∂qf(z) =

(_f₍_z₎₋_f₍_qz₎

z(1−q) , z6= 0,

f0(0), z= 0. (2)

Equivalently (2), may be written as

∂qf(z) = 1 +

∞ X

n=2

[n]qanzn−1, z6= 0

where

[n]q=

1−qn

1−q .

1 _{Department of Mathematics, Yuvaraja’s Collega, University of Mysore, Mysore 570 005, India.}

e-mail: [email protected]; ORCID: https://orcid.org/0000-0001-9243-5885. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-5130-021X. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-1513-8163.

§ Manuscript received: June 26, 2018; accepted: December 20, 2018.

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Note that asq→1, [n]q→n.

Now, recall the definition of the class ofq-starlike functions of order α, 0≤α <1, denoted byS_q∗(α).

Definition 1.1. [2] A function f ∈ Ais said to belong to the class S_q∗(α) if

z∂qf(z)

f(z) −α

1−α −

1 1−q

≤ 1

1−q, z∈ U, (3)

where ∂qf(z) is defined by (2)and 0< q <1.

The following is the equivalent form of Definition 1.1

f ∈S_q∗(α)⇐⇒

z∂qf(z)

f(z) −

1−αq

1−q

≤ 1−α

1−q. (4)

We note that asq→1− the closed disc|ω−(1−q)−1| ≤(1−q)−1 becomes the right-half plane and the classS_q∗(α) reduces toS∗(α), the subclass ofAconsisting of functions which are starlike of orderα(0< α <1) inU. In particular, whenα= 0,the classS_q∗(α) coincides with the classS_q∗:=S_q∗(0), which was first introduced by Ismail et al [4] in 1990 and later it has been considered in [1, 8, 10, 6, 7].

Observe that (3) holds if and only if

z∂qf(z)

f(z) ≺

1 + (1−2α)z

1−qz , (5)

where≺denotes subordination.

Using the definition of the class ofS_q∗(α) and (5) it can be seen that linear transformation

1+(1−2α)z

1−qz maps |z| = r onto the circle with center C(r) =

1+(1−2α)qr

1−q2_r2 and the radius

ρ(r) = (1−₁₋α)(1+_q2_r2q)r.

Thus using subordination principle, we can write

z∂qf(z)

f(z) −

1 + (1−2α)qr2

1−q2_r2

≤ (1−α)(1 +q)r

1−q2_r2 . (6)

Definition 1.2. Let p(z) be analytic inU withp(0) = 0. Then p∈Pm(q, α) if and only if,

P(z) = (m 4 +

1

2)p1(z)−(

m

4 − 1 2)p2(z).

where pj(z)≺ 1+(1−21−qzα)z, j = 1,2, 0< q <1,m≥2.

For m = 2 and α = 0, P2(q) =P(q) consists all functions subordinate to ₁₋1+_qzz, z ∈ U. Also lim_q→1−P(q) =P,the class of functions with positive real part.

Definition 1.3. Let f ∈ A. Then f ∈R∗_q(m, α), if and only if, z∂qf(z)

f(z) ∈Pm(q, α), z∈ U. f in this case, is called a function ofq-bounded radius rotation.

Observe that R∗_q(2,0) =S_q∗ and as q → 1−, α = 0, R∗_q(m, α) =Rm,the class of functions

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2. Main results

We need the following lemmas, to prove our main results.

Lemma 2.1. Let f ∈R∗_q(m, α). Then for m≥2,0< q <1

z∂qf(z)

f(z) −

1 + (1−2α)qr2

1−q2_r2

≤ m

2(1−α)(1 +q)r

1−q2_r2 . (7)

Lemma 2.2. If |u−a| ≤dand |v−a| ≤dwhere a and dare real and a > d≥0, and

ω=u 1

1 +Aeiβ +v

1 1 +A−1e−iβ, where A is real and A >0 and β∈[0, β), then

R(ω)≥a−dsec(β 2).

Lemma 2.3. Let f ∈R∗_q(m, α). Then f ∈S_q∗(α) for |z|< r∗_q(α).where

r_q∗(α) = 4(1−2α)

m(1 +q−2α) +pm2_{(1 +}_q₋₂_α₎2₋₁₆₍₁₋₂_α₎_q. (8)

Proof. Sincef ∈R∗_q(m, α), we have

z∂qf(z)

f(z) =p(z)∈Pm(q, α). This implies thatp(z) can be written as

P(z) = (m 4 +

1

2)p1(z)−(

m

4 − 1 2)p2(z). wherepj(z)≺ 1+(1−2₁₋_qzα)z, j = 1,2, 0< q <1,m≥2.

Therefore

R(z∂qf(z)

f(z) ) =R(p(z))≥(

m

4 + 1 2)(

1+(1−2α)r

1−qr )−( m

4 − 1 2)(

1−(1−2α)r

1+qr )

= 1 +

m

2(1 +q−2α)r+ (1−2α)qr 2

1−q2_r2 ,

and from this, it follows thatR(z∂qf(z)

f(z) )≥0 for|z|< r ∗

q(α).wherer∗q(α) is given by (8).

Observe that as α = 0, f ∈R∗_q(m) and in this case R(z∂qf(z)

f(z) ) >0 for |z|< r ∗

q, where

r_q∗ = 4

m(1+q)+

√

m2₍₁₊_q₎2₋₁₆_q, see [7] and as q → 1

−_, _α _{= 0,} _f _∈ _R

m and in this case

R(zf_f₍0_z(z₎))>0 for |z|< r∗ = 2

m+√m2₋₄, see [3]. Lemma 2.4. Let f ∈R∗_q(m, α). Then

|argf(z)| ≤ m₂(1−α)(1 +q) sin−1r and|argf0(z)| ≤m(1−α)(1 +q) sin−1r.

Theorem 2.1. Let f1, f2 ∈R∗_q(m, α) and let

F(z) =λf1(z) + (1−λ)f2(z), (9)

where 0≤µ= arg₁₋λ_λ < π.Then F ∈S_q∗(α)in |z|< rq,m(α) where rq,m(α) is the smallest positive value ofr satisfying the equation

g(r) = [1 + (1−2α)qr2] cos(µ 2 +

m

2(1−α)(1 +q) sin

−1

r)−m

2(1−α)(1 +q)rsin

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Proof. Usingq-difference operator of (15), we obtained

∂qF(z) =λ∂qf1(z) + (1−λ)∂qf2(z),

and therefore

∂qF(z)

F(z) =

λ∂qf1(z)+(1−λ)∂qf2(z)

λf1(z)+(1−λ)f2(z) = z∂qf1(z)

f1(z) "

1 +

λ

1−λ. f1(z) f2(z)

−1#−1

+z∂qf2(z)

f2(z)

1 +

λ

1−λ. f1(z) f2(z)

−1

. (10)

Put

u= z∂qf2(z)

f2(z) , v=

z∂qf1(z) f1(z) , A=

λ

1−λ. f1(z)

f2(z)

. (11)

From (10) and (17), we obtained

ω(z) = ∂qF(z)

F(z) =u 1

1 +Aeiβ +v

1

1 +A−1e−iβ. (12)

Using Lemma 2.1 and Lemma 2.2, we obtained

R

∂qF(z)

F(z)

≥ 1 + (1−2α)qr

2

1−q2_r2 −

m

2(1−α)(1 +q)r

1−q2_r2 sec( β

2), (13)

where

β= arg₁₋λ_λ.f1(z)

f2(z)

= 2nπ+µ+ argf1(z)−argf2(z).

Now by Lemma 15,

|β| ≤µ+m(1−2α)(1 +q) sin−1r,

and this gives us sec(β₂)≤ 1

cos(µ 2+

m

2(1−α)(1+q) sin −1_r₎. Therefore

Rn∂qF(z)

F(z) o

≥0,if

g(r) = [1 + (1−2α)qr2] cos(µ 2 +

m

2(1−α)(1 +q) sin

−1_r₎₋m

2(1−α)(1 +q)r >0. We note that

g(r) = cos(µ₂),forr = 0, and

g(r) =−m₂(1−α)(1 +q) sin

π−µ m(1−α)(1+q)

<0,whenr = sin

π−µ m(1−α)(1+q)

.

This implies thatg(r) = 0 has a root in the interval

0,sin

π−µ m(1−α)(1+q)

and right hand side of (13) is positive in the disc|z|< rq,m(α),where rq,m(α) is the least positive value of

r satisfying g(r) = 0.

As α= 0, we have the following result, proved by Noor et al [6]. Corollary 2.1. Let f1, f2 ∈R_q∗(m) and let

F(z) =λf1(z) + (1−λ)f2(z), (14)

where 0≤µ= arg₁₋λ_λ < π. Then F ∈S_q∗ in |z|< rq,m where rq,m is the smallest positive value ofr satisfying the equation

g(r) = [1 +qr2] cos(µ 2 +

m

2(1 +q) sin

−1_r₎₋m

2(1 +q)r = 0.

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Corollary 2.2. Let f1, f2 ∈R(m) and let

F(z) =λf1(z) + (1−λ)f2(z), (15)

where 0 ≤ µ = arg₁₋λ_λ < π. Then F ∈ S∗ in |z| < r∗_m where r∗_m is the smallest positive value of r satisfying the equation gm(r) =B(1 +r2)−mr= 0, B= cos µ₂ +msin−1r

.

This gives usr∗_m= m+

√

m2₋₄_B2

2B .As a special case of Corollary 2.2, we takem= 2.Therefore

B =B2= cos µ₂ + 2 sin−1r

,and lim_q→1−R∗_q(2) =S∗.

From these observations, we deduce the radius of starlikeness of linear combination of two

starlike functions is given by r∗₂ = 1−

√

1−B2 2

B2 .

Corollary 2.3. As α = 0, and m = 2. Then, in Theorem 2.1, f1, f2 ∈S_q∗ and it follows that

R

∂qF(z)

F(z)

≥0 in |z|< r_q∗.

where r∗_q is the least positive root of

gq(r) =D1qr2−(1 +q)r+D1= 0,where D1 = cos µ₂ + (1 +q) sin−1r

.

and hence r_q∗= (1+q)−

√

(1+q)2₋₄_qD2 1

2qD1 . Theorem 2.2. Let f1, f2 ∈T

0<q<1S ∗

q(α) and let

F(z) =λf1(z) + (1−λ)f2(z), (16)

where 0≤µ= arg₁₋λ_λ < π. Then F maps the disc |z|< rµ onto a convex domain, where

rµ is the least positive value ofr that satisfies the equation

gµ(r) =Dr2−2r1r+Dr12, where r1 = 2− √

3+α2

1+α , D= cos( µ

2 + 2(1−α) sin −1₍r

r1)).

Proof. It has been shown in [2] that

\

0<q<1

S_q∗(α) =S∗(α).

It is well known [9] that f ∈ S∗(α) is convex of order α in the disc |z|< r1 = 2−

√ 3+α2

1+α .

with these facts, we proceed to find the radius of convexity for the functionF following the technique used in Theorem 2.1.

We can write

1+zF

00₍_z₎ F0₍_z₎ =

1 +zf

00 1(z) f₁0(z)

"

1 +

λ

1−λ. f₁0(z)

f₂0(z)

−1#−1

+

1 +zf

00 2(z) f₂0(z) 1 +

λ

1−λ. f1(z)

f2(z)

−1 .

Put

u=

1 +zf

00 1(z) f₁0(z)

, v=

1 +zf

00 2(z) f₂0(z)

, A=

λ

1−λ. f₁0(z)

f₂0(z)

, β= arg

λ

1−λ. f₁0(z)

f₂0(z)

. (17)

Now, for r1 = 2−

√ 3+α2

1+α ,we have

u−r

2 1 +r2 r2

1 −r2

≤ 2rr1

r2 1−r2

,

v− r

2 1+r2 r2₁−r2

≤ 2rr1

r2₁−r2.

Therefore

ω(z) = 1 +zF

00₍_z₎ F0₍_z₎ =u

1

1 +Aeiβ +v

1

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where

β= arg₁₋λ_λ.f10(z)

f0 2(z)

= 2nπ+µ+ argf₁0(z)−argf₂0(z),and so

|β| ≤µ+ 4(1−α) sin−1(_r₋₁r ),

Therefore

Rn1 +zF_F000₍(_zz₎)

o

> 0, if Tµ(r) = (r21 +r2) cos(

µ

2 + 2(1−α) sin −1₍ r

r−1)−2r1r = 0, where r1 = 2−

√ 3+α2

1+α .

That is

Tµ(r) =Dr2−2r1r+Dr21, D= cos(

µ

2 + 2(1−α) sin −1₍r

r1)). Hence

rµ=

r1− p

r2

1−D2r21

D . (19)

Hence F maps the disc |z| < rµ onto a convex of order α domain, where rµ is given by

(19).

Remark Asα= 0 Theorem 2.1 reduces to Theorem 2 in [6].

Acknowledgement Our thanks are due to the anonymous referee for careful reading and constructive suggestions for the improvement in the first draft of this paper.

References

[1] Agrawal, S. and Sahoo, S. K., (2014). Geometric properties of basic hypergeometric functions, Difference Equ. Appl., 20(11), pp. 1502-1522.

[2] Agrawal, S. and Sahoo, S. K., (2017). A generalization of starlike functions of order alpha, Hokkaido Math. J., 46(1), pp. 15-27.

[3] Goodman, A. W., (1983). Univalent Functions, Vol I. Washington, New Jersey: Polygonal Publishing House.

[4] Ismail, M. E. H., Merkes, E., Styer, D. (1990). A generalization of starlike functions, Complex Variables, 14, pp. 77-84.

[5] Jackson, F. H., (1909). Onq-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46, pp. 253-281.

[6] Noor, K. I. and Noor, M. A., (2017). Linear combinations of generalized q-starlike functions, Appl. Math. Infor. Sci., 11(3), pp. 745-748.

[7] Noor, K. I., and Riaz, S., (2017). Generalized q-starlike functions, Studia Scientiarum Mathematicarum Hungarica, 54(4) , pp. 509-522.

[8] Raghavendar, K. and Swaminathan, A., (2012). Close-to-convexity of basic hypergeometric functions using their Taylor coefficients, J. Math. Appl., 35 , pp. 111-125.

[9] Robertson, M. I., (1936). On the theory of univalent functions, Annals Math., pp. 374-408.

[10] Sahoo, S. K. and Sharma, N. L., (2015). On a generalization of close-to-convex functions, Ann. Polon. Math., 113(1), pp. 93-108.

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Read S.A. Qahtanwas born in Yemen. He got his B. Sc. degree in mathematics in 2002 from Dhamar University, Yemen. He got his M. Sc. degree from University of Mysore, India. He is right now a Ph.D. student at University of Mysore, India.