The Fekete Szegö inequality for close to convex functions with respect to a certain starlike function dependent on a real parameter

R E S E A R C H

Open Access

The Fekete-Szegö inequality

for close-to-convex functions with

respect to a certain starlike function

dependent on a real parameter

Bogumiła Kowalczyk

_{and Adam Lecko}

*_{Correspondence:} [email protected] 2_{Department of Analysis and} Differential Equations, University of Warmia and Mazury, Słoneczna 54, Olsztyn, 10-710, Poland

Full list of author information is available at the end of the article

Abstract

Given

α

αz)

normalized functionfinDis calledclose-to-convex with respect to gαif there exists

δ

π/2) such that

Re

eiδzf

_(z)

gα(z)

> 0, z∈D.

For the classC(gα) of all close-to-convex functions with respect togα, the

Fekete-Szegö problem is studied.

MSC: Primary 30C45

Keywords: Fekete-Szegö problem; close-to-convex functions; close-to-convex functions with respect to the Koebe function; close-to-convex functions with argument

δ; functions convex in the positive direction of the imaginary axis

1 Introduction

The classical problem settled by Fekete and Szegö [] is to find for eachλ∈[, ] the max-imum value of the coefficient functional

λ(f) :=a–λa

over the classSof univalent functionsf in the unit diskD:={z∈C:|z|< }of the form

f(z) =z+

∞

n=

anzn, z∈D. (.)

By applying the Loewner method they proved that

max

f∈S λ(f) = ⎧ ⎨ ⎩

 + exp(–λ/( –λ)), λ∈[, ),

, λ= .

The problem of calculatingmaxf∈Fλ(f) for various compact subclassesFof the class Aof all analytic functionsf inDof the form (.), as well as forλbeing an arbitrary real or complex number, was considered by many authors (see,e.g., [–]).

LetS∗ denote the class ofstarlikefunctions,i.e., the class of all functionsf ∈Asuch that

Rezf

_(z)

f(z) > , z∈D. (.)

Givenδ∈(–π/,π/) andg∈S∗, letCδ(g) denote the class of functions called close-to-convex with argumentδwith respect to g,i.e., the class of all functionsf ∈Asuch that

Re eiδzf

_(z)

g(z)

> , z∈D. (.)

Let

C(g) := δ∈(–π/,π/)

Cδ(g), Cδ:=

g∈S∗

Cδ(g)

denote the classes of functions calledclose-to-convex with respect to gandclose-to-convex with argumentδ, respectively, and let

C:=

δ∈(–π/,π/)

Cδ=

δ∈(–π/,π/)

g∈S∗

Cδ(g)

denote the class ofclose-to-convexfunctions (see [, pp.-], [, ]). It is well known thatS∗andCare the subclasses ofS.

By using a specific starlike functiong, inequality (.) defines the related classCδ(g). Givenα∈[, ], let

gα(z) :=

z

( –αz) = ∞

n=

nαn–zn, z∈D. (.)

It is easy to check that eachgαsatisfies (.),i.e.,gα∈S∗for everyα∈[, ]. Then (.) is of the form

Reeiδ_{( –αz)}_{f(z)> , z∈D, (.)}

and defines the classCδ(gα), and further the classC(gα). Such classes of functions were studied in [, ] and [], where some generalization of the Robertson condition for convexity in one direction [] was discussed.

Note that forα:=  we get the Koebe functiong=:k. Then condition (.) defines the

classCδ(k) and further the classC(k) of functionsclose-to-convex with respect to the Koebe function. Such functions have a well-known geometrical meaning, namely condition (.) geometrically says that the functionh:= eiδ_{f has the boundary normalization}

lim

t→∞h

andh(D) is a domain such that{w+t:t≥} ⊂h(D) for everyw∈h(D). Such functions

h, clearly univalent as close-to-convex, and domainsh(D) are calledconvex in the positive (negative) direction of the real axisand are related to functionsconvex in the direction of the imaginary axis(see,e.g., [–], [, Chapter VI], []).

Forα:=  we have the identity functiong(z) =z,z∈D, and then condition (.) is of

the form

Reeiδf(z)> , z∈D. (.)

Functions f having such a property are calledof bounded turning with argumentδ and form the classCδ(g) denoted usually asP(δ). Functions in the classP:=C(g) are usually

calledof bounded turning (cf. [, Vol. I, p.]). On the other hand, condition (.) is known as the famous criterium of univalence due to Noshiro [] and Warschawski [] (cf.[, Vol. I, p.]). In this way condition (.) creates a simple parametric passage from the classP(δ) to the classCδ(k).

The main goal of this paper is to study the Fekete-Szegö problem for the classesC(gα), α∈[, ]. For the classC(k),i.e., forα= , the Fekete-Szegö problem was examined in [], where it was shown that

max

f∈C(k)λ(f)≤ ⎧ ⎨ ⎩

| – λ|, λ∈(–∞, /]∪[, +∞),

 ·

(–λ)

–|–λ|+| –λ|+



, λ∈[/, ],

with sharpness of the result forλ∈R\(/, ). Recall here that in [] Keogh and Merkes proved that

max

f∈C λ(f) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

| – λ|, λ∈(–∞, /]∪[, +∞), / + /(λ), λ∈[/, /],

, λ∈[/, ].

Forλ∈[, ] Koepf in [] extended the above result for the class of close-to-convex func-tions showing that

max

f∈C λ(f) =maxf∈C λ(f).

For other results on the Fekete-Szegö problem for various subclasses of close-to-convex functions, particularly forstrongly close-to-convexfunctions, see [–] and [].

For the classPandλ∈[, ], we get the following sharp result published, among other results, in [, Theorem .], namely

max

f∈P

λ(f) =  .

2 Main result

ByPwe denote the class of all analytic functionspinDof the form

p(z) =  +

∞

n=

having a positive real part inD. For eachε∈T:={z∈C:|z|= }, let

Lε(z) :=  +εz

 –εz, z∈D,L:=L.

Inequalities (.) and (.) below are well known (see,e.g., [, pp., ]).

Lemma . If p∈Pis of the form(.),then

|cn| ≤, n∈N, (.)

and c–

c 



≤ –|c|



 . (.)

Both inequalities are sharp.The equality in(.)holds for every function Lε,ε∈T.The

equality in(.)holds for every function

pt,θ(z) :=tL

eiθ_{z+ ( –t)Le}iθ_z

=  + teiθ

z+ eiθ

z+· · ·, z∈D, (.)

where t∈[, ]andθ∈R.

Now we prove the main theorem of this paper. The source of the method of proof is in Koepf ’s paper [], where the upper bound ofλfor close-to-convex functions withλ restricted to the interval (/, /) was calculated. However, we modify this technique and use it homogeneously for the classC(gα) for all realλ, partially analogously as in [] for the classC(k), and in []. We apply also the powerful Laguerre’s rule of counting zeros of polynomials in an interval. We propose Laguerre’s algorithm for such a computation by its simplicity, usefulness and efficiency.

We shortly recall Laguerre’s rule of counting zeros of polynomials in an interval (see [, ], [, pp.-]). Given a real polynomial

Q(u) =aun+aun–+· · ·+an–u+an, (.)

consider a finite sequence (qk),k= , , . . . ,n, of polynomials of the form

qk(u) = k

j=

ajuk–j. (.)

For eachu∈R, letN(Q;u) denote the number of sign changes in the sequence (qk(u)), k= , , . . . ,n. Given an intervalI⊂R, denote byZ(Q;I) the number of zeros ofQinI

counted with their orders. Then the following theorem due to Laguerre holds.

Note thatqk() =akandqk() = k

j=aj. Thus, when (a,b) := (, ), Theorem . reduces

to the following useful corollary.

Corollary . If Q()= and Q()= ,then Z(Q; (, )) =N(Q; ) –N(Q; )or N(Q; ) –

N(Q; ) –Z(Q; (, ))is an even positive integer,where N(Q; )and N(Q; )are the numbers of sign changes in the sequence of polynomial coefficients(ak)and in the sequence of sums

(k_j=aj),with k= , , . . . ,n,respectively.

The main theorem of the paper is as follows.

Theorem . Letα∈[, ].Then

max

f∈C(gα)

λ(f)

≤ ⎧ ⎨ ⎩ |

+  α+α

_{– ( +α)}_{λ|, λ∈R\(τ}

(α),τ(α)), 

+α(  ·

(–λ)

–|–λ|+| –λ|), λ∈[τ(α),τ(α)],

(.)

where

τ(α) :=

α

( +α), τ(α) :=

( +α) ( +α).

For eachα∈(, ]and eachλ∈R\(/,τ(α)),as well as forα:= and eachλ∈R,the inequality is sharp and the equality is attained by a function inC(gα).In particular,

(i)whenα∈(, ],for eachλ∈[τ(α), /]the second equality in(.)is attained by the function fα,tα,λgiven by the differential equation

fα,tα,λ(z) =

ptα,λ,(z)

( –αz), fα,tα,λ() := ,z∈D, (.)

where tα,λ:=α(/(λ) – );

(ii)whenα∈(, ],for eachλ∈R\(τ(α),τ(α))the first equality in(.)is attained by the function fα,,given by(.)with tα,λ≡,i.e.,whenα∈(, ),by the function

fα,(z) =

 ( –α)log

 –αz

 –z –

 +α  –α ·

z

 –αz, z∈D,log := , (.)

and whenα= ,by the Koebe function f,:=k;

(iii)whenα= ,for eachλ∈[, /]the second equality in(.)is attained by the function

f,(z) := –z+log

 +z

 –z, z∈D,log := ; (.)

and for eachλ∈R\(, /)the first equality in(.)is attained by the function

f,(z) := –z– log( –z), z∈D,log := . (.)

Proof Fixα∈[, ]. Observe from (.) thatf∈C(gα) if and only if

for someδ∈(–π/,π/) andp∈P. Thus

zf(z) = e–iδgα(z)

p(z)cosδ+ isinδ, z∈D. (.)

Setting the series (.), (.) and (.) into (.), by comparing coefficients, we get

a=

 

ce–iδcosδ+ α

(.)

and

a=

 

ce–iδcosδ+ αce–iδcosδ+ α

. (.)

Letλ∈R. Using (.), from (.) and (.), we have

λ(f) =a–λa

= ce

–iδ_cosδ +

αce

–iδ_cosδ +α

– λ

c_e–iδ_cos_{δ+ αc}

e–iδcosδ+ α

=α( –λ) + 

c–

c_



e–iδcosδ

+c     – λe

–iδ_cosδ

e–iδcosδ+α

  –λ

ce–iδcosδ

≤α| –λ|+ 

 –|c|





cosδ

+|c|





 –

λe

–iδ_cosδcosδ +α

–λ

|c|cosδ =α| –λ|

+

 +

|c|



 –

λ– λ



cos_{δ– }

+α –λ

|c|

cosδ. (.)

Setx:=|c|andy:=cosδ. Clearly,y∈(, ] and, in view of (.),x∈[, ]. It is convenient

to useγ :=  – λinstead ofλin further computation. Forγ ∈R, let

sγ(y) :=

 –  – γ 

y_{, y∈[, ].}

SetR:= [, ]×[, ]. Forα∈[, ] andγ ∈R, define

Fα,γ(x,y) :=  α

_{| +γ|+}



 +x





sγ(y) – 

+α|γ|x

y, (x,y)∈R.

Consequently, in view of (.) we have

max

f∈C(gα)

λ(f)≤ max

Now, for eachα∈[, ] andγ ∈R, we find the maximum value ofFα,γon the rectangleR. . In the corners ofRwe have

Fα,γ(, ) =Fα,γ(, ) =  α

_{| +γ|, (.)}

Fα,γ(, ) =  α

_{| +γ|+}

, (.)

Fα,γ(, ) =  α

_{| +γ|+}

( + α)|γ|. (.)

. Forx=  andy∈(, ) we have a linear function and forx∈(, ) andy=  we have a constant function.

. Forx∈(, ) andy= , let

Gα,γ(x) :=Fα,γ(x, ) =  

 

|γ|– x+α|γ|x+α| +γ|+ 

.

For|γ|=  we get the linear functions, so let|γ| = . ThenGα,γ(x) =  if and only if

x= α|γ|  –|γ|=:xα,γ.

Thusxα,γ∈(, ) if and only if

α= ∧ <|γ|< 

 +α. (.)

Moreover, we have

Fα,γ(xα,γ, ) =  

α_γ

 –|γ|+α

_{| +γ|+ }

. (.)

. Forx=  andy∈(, ), let

Hα,γ(y) :=Fα,γ(,y) =  

α| +γ|+ ysγ(y) + α|γ|y

.

For|γ|=  we have the linear functions, evidently, so let|γ| = . Note first that

sγ(y) > , y∈(, ). (.)

Taking into account (.), we have

ysγ(y) =

–( –_γ)y

 – ( – _γ_)y

=s



γ(y) – 

sγ(y)

, y∈(, ). (.)

Using (.) we get

Hα,γ(y) =  

sγ(y) +

s

γ(y) – 

sγ(y)

+α|γ|

if and only if

sγ(y) +α|γ|sγ(y) –  = , y∈(, ),

i.e., in view of (.) if and only if

sγ(y) =

–α|γ|+ +α_γ

 =:sα,γ, y∈(, ).

Since|γ| = , so from the above we get the equation

y= –α

_γ_{+α|γ| +α}_γ

( –γ₎ , y∈(, ). (.)

Thus the solution of equation (.), and hence of (.), exists if and only if

 < –α

_γ_{+α|γ| +α}_γ

( –γ₎ < . (.)

Elementary computing shows that (.) holds if and only if

|γ|<



 +α. (.)

Thus the functionHα,γ has a critical point in (, ), namely

y=

 –α_γ_+α|_γ|_α_γ_{+ }

( –γ₎ =:yα,γ,

as the unique solution of (.), if and only if (.) holds. Moreover,

Fα,γ(,yα,γ) =  α

_{| +γ|}

+ 

 –α_γ_+α|_γ|_{ +α}_γ

( –γ₎

 +α_γ_{+ α|γ|. (.)}

. We will prove that for eachα∈[, ] andγ ∈Rthe functionFα,γ has no critical point in (, )×(, ).

Sincey=  andx= , we have

∂Fα,γ ∂x = 

if and only if

sγ(y) =  – α|γ|

x , y∈(, ). (.)

Sincex> , by comparing (.) and (.), we see thatx>α|γ|. By a simple observation, we deduce that the solution of (.) can exist only when

Squaring then (.), we obtain

sγ(y) –  = – α|γ|

x +

α_γ

x . (.)

Since by (.),sγ(y)=  fory∈(, ), taking into account (.), we have

∂Fα,γ ∂y =

 +

x



sγ(y) – 

+ α|γ|x+

x

 ·

s

γ(y) – 

sγ(y) .

Thus, by using (.) and (.), after simplifying we have

∂Fα,γ ∂y = 

if and only if

α|γ|x– x+ α|γ|= , x∈(, ). (.)

It follows at once that forα=  andγ ∈R, as well as forα∈(, ] and|γ| ≥/α, equation (.) has no root. Thus by (.) we consider

α= ∧ |γ| = ∧ <|γ|< /α∧x>α|γ|. (.)

Solving now (.), we have

x= – 

 –α_γ

α|γ| =:x;α,γ, x=

 +  –α_γ

α|γ| =:x;α,γ.

Clearly,x;α,γ ∈/(, ) and it remains to considerx;α,γ. It is easy to check thatx;α,γ>α|γ|. Thus settingx:=x;α,γ into (.) and computing, we have

y=

α|γ|

x;α,γ –

αγ

x_;α,γ

 –_γ =

( –αγ–  –α_γ_)α_γ

( – –α_γ₎_{( –γ}₎ . (.)

A solution in (, ) of (.) exists if and only if

 <( –α

_γ_{–  –α}_γ_)α_γ

( – –α_γ₎_{( –γ}₎ < . (.)

By (.) consider

α= ∧ |γ| = ∧ <|γ|< 

α. (.)

We prove that then condition (.) is false. When  <|γ|< /α, then an easy computation shows that the left-hand inequality in (.) is false. Thus by (.) it remains to consider

α= ∧ <|γ|< 

By an easy computation we check that then the left-hand inequality in (.) holds. Since  –γ_{> , write the right-hand inequality in (.) as}

 –  + αγ –α_γ_<α_+α_γ_{–  + α}_γ_{+ .}

The last step is to show, which does not cause difficulties, that under the assumption (.) the above inequality is false. We omit the details.

Summarizing, we proved that condition (.) is false, so equation (.) has no solution in (, ).

Thus the proof that forα∈[, ] andγ ∈Rthe functionFα,γ has no critical point in (, )×(, ) is finished.

. Now we calculate the maximum value ofFα,γ inR, which, as was shown, is attained on the boundary ofR. Letα∈[, ]. Taking into account Part  with (.) and Part  with (.), we consider the following cases.

(A)|γ| ≥/( +α). Then the maximum value ofFα,γ is attained in a corner ofR. Thus by (.)-(.) an easy computation shows that

max

(x,y)∈RFα,γ(x,y) =Fα,γ(, ) =

 α

_{+ ( +α)}_{γ. (.)}

(B)√/( +α)≤ |γ|< /(α+ ). Then the maximum value ofFα,γ is attained in a corner ofRor in (xα,γ, ). Thus, by (.)-(.) and (.), we calculate that

max

(x,y)∈RFα,γ(x,y) =Fα,γ(xα,γ, ) =

  ·

αγ  –|γ|+

 α

_{| +γ|+}

. (.)

(C)γ = . Then the maximum value ofFα,is attained in a corner ofRor in the point

(,yα,) = (, / √

). Thus, by (.)-(.) and (.) withγ:= , we see that

max

(x,y)∈RFα,(x,y) =Fα,(, ) =

 α

₊

. (.)

(D)  <|γ|<√/(α+ ). Then we compare all values (.)-(.), and by (.) and (.),Fα,γ(xα,γ, ) andFα,γ(,yα,γ). We will show that the valueFα,γ(xα,γ, ) is the largest one. As it is easy to observe, it is enough to prove that

Fα,γ(xα,γ, )≥Fα,γ(,yα,γ) (.)

i.e., in view of (.) and (.), after a simple computation, we have



 + α

_γ

 –|γ|

≥

 –α_γ_+α|_γ|_{ +α}_γ

( –γ₎

 +α_γ_{+ α|γ|. (.)}

As|γ|< , so squaring (.) and computing, we equivalently have

α|γ|+α– αγ+α+ |γ|–α+ γ– |γ|+ 

To verify that (.) holds, settinguα:=

√

/(α+ ), we will show that for everyα∈[, ] we have

Qα(u) :=αu+

α– αu+α+ u–α+ u– u+ 

≥αu( –u)αu+ /=:Sα(u), u∈[,uα]. (.)

(o) Forα= , inequality (.) is evidently true.

(o) Forα= , we haveu=  and inequality (.) after computing is equivalent to the

evidently true inequality

(u– )u+ u+ u– u+ u+ ≥, u∈[, ].

(o_{) Letα∈(, ). We will show that foru∈[,u}

α],

Vα(u) :=Qα(u) –Sα(u) =

Qα(u) –Sα(u)

Qα(u) +Sα(u)

> . (.)

Further, taking into account thatQαandSαare continuous functions with

Qα() –Sα() =  > ,

from (.) we deduce that

Qα(u) –Sα(u) > , u∈[,uα],

which confirms (.) and further (.).

Now we prove that (.) holds,i.e., after computation we have

Vα(u) =

α–αu+α– α+ αu

+α– α– αu+–α+ α+ α+ u

+α– α– u–α+ α+ u

+α+ u–α+ u– u+ 

> , u∈[,uα].

As in (.), let (qk),k= , , . . . , , be a sequence of polynomials of the form

qk(u) = k

j=

ajuk–j, u∈[,uα],

corresponding to the polynomialQ:=Vαin (.) for Laguerre’s rule in [,uα].

(a) First we check the signs of the elements of the sequence (qk()),i.e., of the sequence

(ak) fork= , , . . . , . A simple computation shows that forα∈(, ) we haveq() < , q() > , q() < , q() > , q() < , q() < , q() > , q() < , q() <  and q() > . Hence

(b) Now we check the signs of the elements of the sequence (qk(uα)) fork= , , . . . , . After the detailed computation and arguments based on Laguerre’s rule, we show that

q(uα) < ,q(uα) > ,q(uα) < ,q(uα) > ,q(uα) < ,q(uα) > ,q(uα) > ,q(uα) <  andq(uα) > . Moreover, we show that there exists a uniqueα∈(, ) such thatq(uα) =  andq(uα) <  forα∈(,α) andq(uα) >  forα∈(α, ). Thus for three cases, namely

forα∈(,α),α:=αandα∈(α, ), we have

N(Vα;uα) = .

Hence, by (.) and by Corollary ., we conclude that for eachα∈(, ) the polynomial

Vαhas no zero in (,uα), and sinceVα() =  > , so (.) holds.

Now we shortly explain the method of describing the signs ofqk(uα),k= , . . . , . Note that the casek=  is evident since

q(uα) =α

α– < , α∈(, ).

For other cases,i.e., fork= , . . . , , we use Laguerre’s rule in each case in the same manner. We clarify this for the casek= . We have

q(u) =

α–αu+α– α+ αu+ α– α– α, u∈[,uα].

We will show that

q(uα) < , α∈(, ), (.)

i.e., after computing we get

√

αα– α+ <–α– α+ α+ √α+ , α∈(, ). (.)

To verify that (.) holds, we will show that

r(t) :=√tt– t+ 

<–t– t+ t+ √t+  =:s(t), t∈[, ]. (.)

Applying Corollary ., we see that

w(t) :=s(t) –r(t)

= –t+ t+ t+ t+ t– t– t

– t– t+ t+ t+ t+ 

=:



j=

bjt–j> , t∈[, ]. (.)

Indeed, the numbers of sign changes in the sequence of polynomial coefficients (bk) and

in the sequence of sums (k_j=bj), wherek= , , . . . , , equal ,i.e.,N(w; ) =N(w; ) = .

 > , so (.) holds. Hence, and by the fact thats() >r(), we deduce that (.) holds, which confirms (.).

Note here that for the casek=  we show that the equation

q(uα) = 

has a unique solutionα=:α∈(, ). In this case, we show by using Laguerre’s rule that

the corresponding polynomialwas in (.) has a unique zero in (, ) and further we deduce that

q(uα) < , α∈(,α)

and

q(uα) > , α∈(α, ).

Summarizing, taking into account (.)-(.), we have

max

(x,y)∈RFα,γ(x,y) = ⎧ ⎨ ⎩  |α

_{+ ( +α)}_{γ|, |γ| ≥}  +α,

 ·

αγ

–|γ|+



α| +γ|+ 

, |γ| ≤  +α.

Finally, substitutingγ =  – λ, the above yields (.).

Now we deal with the sharpness of the result. Let α∈(, ]. We prove that for each λ∈(–∞, /]∪[τ(α), +∞) inequality (.) is sharp. Letλ∈[τ(α), /]. Since then

α

 ·

( – λ)

 –| – λ|+| –λ|

+ =α



 λ–

 

+ ,

inequality (.) is of the form

max

f∈C(gα)

λ(f)≤α

 λ–

 

+ , λ∈

τ(α), /

. (.)

Lettα,λ:=α(/(λ) – ). Thentα,λ∈[, ] and, in view of (.),ptα,λ,∈P, withc= tα,λand

c= . Settingδ:=  andp:=ptα,λ,into (.), we get the functionfα,tα,λgiven by equation

(.) for which, by (.) and (.),

a=tα,λ+α= α

λ, a=  

 + αtα,λ+ α

=  +α



 λ–

 

.

Hence

λ(fα,tα,λ) =

 +α



 λ–

 

,

which makes the equality in (.), so in (.). Clearly, fα,tα,λ∈C(gα) because (.) is

Letλ∈(–∞,τ(α)]∪[τ(α), +∞). Then inequality (.) is of the form

max

f∈C(gα)

λ(f)≤

_ +

α+α

_{– ( +α)}_{λ. (.)}

Setδ:=  andp:=Linto (.). Then forα∈(, ) we get the functionfα,given by (.)

and forα=  we get the Koebe functionf,=k, with

a=  +α, a=

 

 + α+ α, α∈(, ].

Hence

λ(fα,) = _+

α+α

_{– ( +α)}_λ,

which makes the equality in (.), so in (.). Clearly,fα,∈C(gα) forα∈(, ].

Letα:= . We prove that for eachλ∈Rinequality (.) is sharp. Forλ∈[τ(),τ()] =

[, /] inequality (.) is of the form

max

f∈Pλ(f)≤



. (.)

Settingδ:=  and, by (.),p:=p,into (.), we get the functionf,given by (.) with a=  anda= /. Hence

λ(f,) =

 ,

which makes the equality in (.), so in (.).

Forλ∈(–∞, ]∪[/, +∞) inequality (.) is of the form

max

f∈P

λ(f)≤

_ –λ. (.)

Settingδ:=  andp:=Linto (.), we get the functionf,given by (.) witha=  and a= /. Hence

λ(f,) = _–λ,

which makes the equality in (.), so in (.).

Remark . Particularly, from (.) forα∈[, ] we have

max

f∈C(gα)

λ(f) =

⎧ ⎨ ⎩ |

+  α+α

_{– ( +α)}_{λ|, λ∈R\(τ}

(α),τ(α)), 

+α(  λ–



), λ∈[τ(α), /].

Forα:=  we haveτ() = / andτ() = ,g=k, and then Theorem . reduces to the

Corollary .

max

f∈C(k)λ(f)≤ ⎧ ⎨ ⎩

| – λ|, λ∈(–∞, /]∪[, +∞),

 +

 ·

(–λ)

–|–λ|+| –λ|, λ∈[/, ].

(.)

For each λ∈(–∞, /]∪[, +∞),the inequality is sharp and the equality is attained by a function inC(k).In particular,for eachλ∈[/, /]the second equality in(.)is attained by the function ftλ:=f,t,λgiven by differential equation(.),where tλ:=t,λ.For

eachλ∈(–∞, /]∪[, +∞),the first equality in(.)is attained by the Koebe function f:=k.

Forα:=  we haveτ() =  andτ() = /, and Theorem . yields the following.

Corollary .

max

f∈Pλ(f) = ⎧ ⎨ ⎩ |

–λ|, λ∈(–∞, ]∪[/, +∞), 

, λ∈[, /].

(.)

For eachλ∈[, /]the second equality in(.)is attained by the function given by

(.).For eachλ∈(–∞, ]∪[/, +∞),the first equality in(.)is attained by function

(.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Department of Applied Mathematics, University of Warmia and Mazury, Słoneczna 54, Olsztyn, 10-710, Poland.} 2_{Department of Analysis and Differential Equations, University of Warmia and Mazury, Słoneczna 54, Olsztyn, 10-710,} Poland.

Received: 9 December 2013 Accepted: 31 January 2014 Published:13 Feb 2014 References

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10.1186/1029-242X-2014-65

Cite this article as:Kowalczyk and Lecko:The Fekete-Szegö inequality for close-to-convex functions with respect to a certain starlike function dependent on a real parameter.Journal of Inequalities and Applications