SUBCLASSES OF UNIFORMLY CONVEX FUNCTIONS [627590]

SUBCLASSES OF UNIFORMLY CONVEX FUNCTIONS
OBTAINED BY USING AN INTEGRAL OPERATOR AND THE
THEORY OF STRONG DIFFERENTIAL SUBORDINATIONS
ROXANA S ENDRUT  IU1
Abstract. The concept of dierential subordination was introduced in [6] by
S.S.Miller and P.T.Mocanu and the concept of strong dierential subordina-
tion was introduced in [2] by J.A.Antonino and S.Romaquera. In this paper
we dene some subclasses of uniformly convex functions with respect to a
convex domain included in the right half plane D, obtained by using an integral
operator and the theory of strong dierential subordinations.
1.Introduction
LetUdenote the unit disc of the complex plane :
U=fz2C:jzj<1g
and
U=fz2C:jzj1g:
LetH(UU) denote the class of analytic functions in UU.
In [10], the authors have dened the class
H[a;n] =ff2H(UU) :f(z;) =a+an()zn+an+1()zn+1+


; z2U;2Ug
withak() holomorphic functions in U,kn,
Hu(U) =ff2H[a;n] :f(z;) univalent in UU;for allz2U;2Ug;
An=ff2H[a;n] :f(z;) =z+a2()z2+


+an()zn+


; z2U;2Ug
withA1=A;
and
S=ff2An:f(z;) univalent in UU;for allz2U;2Ug:
Let
S=
f2A: Rezf0(z;)
f(z;)>0; z2U;for all2U
denote the class of starlike functions in UU,
K=
f2A: Rezf00(z;)
f0(z;)+ 1>0; z2U;for all2U
2000 Mathematics Subject Classication. Primary 30C80, Secondary 30C45, 30A20.
Key words and phrases. integral operator, dierential subordination, strong dierential sub-
ordination, univalent function, convex function.
1

2 R. S ENDRUT  IU
denote the class of normalized convex functions in UU, and
C=
f2A:9'2K;Ref0(z;)
'0(z;)>0; z2U;for all2U
denote the class of close-to-convex functions in UU.
Denition 1.1. [10] LetH(z;) andf(z;) be analytic in UU. The function
f(z;) is said to be strongly subordinate to H(z;), orH(z;) is said to be
strongly superordinate to f(z;), if there exists a function !analytic in Uwith
!(0) = 0,j!(z)j<1, such that f(z;) =H[!(z);], for all2U. In such a case
we writef(z;)H(z;); z2U;2U:
Remark 1.2.(i) IfH(z;) is analytic in UUand univalent in Ufor all2U,
Denition 1.1 is equivalent to f(0;) =H[0;], for all2Uand
f(UU)H(UU):
(ii) IfH(z;)H(z) andf(z;)f(z) then the strong subordination becomes
the usual notion of subordination.
Denition 1.3. [3] Let consider the integral operator La:An!Andened
as:
f(z;) =LaF(z;) =1 +a
zaZz
0F(t;)ta1dt; a2C;Rea0: (1.1)
In the case a= 1;2;3;


this operator was introduced by S.D.Bernardi [3]
and it was studied by many authors.
Denition 1.4. [12] Forf(z;)2An,n2N[f0g, we dene the dierential
operator:In:An!An
I0f(z;) =f(z;)
I1f(z;) =zf0(z;)




In+1f(z;) =z[Inf(z;)]0(z2U;2U):
We note that the derivative is to respect to the rst variable.
Property 1.5. Forf(z;)2An,n2N[f0g, with the dierential operator
In:An!Anwe have:
z[In+1f(z;)]0=Inf(z;) (z2U;2U):

SUBCLASSES OF UNIFORMLY CONVEX FUNCTIONS 3
2.Preliminary results
The next two denitions (given in [5]) are adapted to the class H[a;n]:
Denition 2.1. [5] Let2[0;1] andf(z;)2 An. We say that fis a
uniformly convex function if
Ref(1)zf0(z;)
f(z;)+(1+zf00(z;)
f0(z;))gj (1)(zf0(z;)
f(z;)1)+zf00(z;)
f0(z;)j;(z2U;2U):
We denote this class by UM.
Remark 2.2.Geometric interpretation: f2UMif and only if
J(;f;z;) = (1)zf0(z;)
f(z;)+(1 +zf00(z;)
f0(z;))
take all values in the parabolic region
= fw:jw1jRewg=fw=u+{v:
v22u1g:For a xed we obtainUM 0=SP, where the class SPwas
introduced by F. Ronning in [11] and UMM, whereMis the well know
class ofconvex functions introduced by P.T. Mocanu in [9].
Denition 2.3. [5] Let2[0;1] andn2N. We say that f(z;)2Anis in
the classUDn;(;
);0;
2[1;1);+
0 if
Re [(1)In+1f(z;)
Inf(z;)+In+2f(z;)
In+1f(z;)]j(1)In+1f(z;)
Inf(z;)+In+2f(z;)
In+1f(z;)1j+
:
Remark 2.4.Geometric interpretation: f2UDn;(;
) if and only if
Jn(;f;z;) = (1)In+1f(z;)
Inf(z;)+In+2f(z;)
In+1f(z;)
takes all values in the convex domain included in right half plane D;
, where
D;
is a elliptic region for  > 1, a parabolic region for = 1, a hyperbolic
region for 0 < < 1, the half plane for = 0. We have UD 0;(1;0) =UM.
The next theorem is a result due to so called "admissible functions method"
introduced by P.T. Mocanu and S.S. Miller (see [6], [7], [8]) and adapted to the
classH[a;n].
Theorem 2.5. [6],[7],[8]Leth2KandRe [h(z;) +]>0,z2U;2U.
Ifp2H(UU)withp(0;) =h(0;)andpsatises the Briot-Bouquet strong
dierential subordination
p(z;) +zp0(z;)
p(z;) +h(z;);

4 R. S ENDRUT  IU
thenp(z;)h(z;):
The next denition (given in [4]) is adapted to the class H[a;n].
Denition 2.6. [4] The function f(z;)2Anis n-starlike with respect to con-
vex domain included in right half plane Dif the dierential expressionIn+1f(z;)
Inf(z;)
takes values in the domain D.
Remark 2.7.If we consider q(z;) an univalent function with q(0;) = 1, Req(z;)>
0,q0(0;)>0 which maps the unit disc Uinto the convex domain Dwe have:
In+1f(z;)
Inf(z;)q(z;):
We denote by Sn(q) the class of all these functions.
3.Main results
Letq(z;) be an univalent function with q(0;) = 1,q0(0;)>0, which maps
the unit disc Uinto a convex domain included in right half plane D.
The next denition (given in [1]) is adapted to the class H[a;n].
Denition 3.1. [1] Letf(z;)2 Anand2[0;1]. We say that fis a
uniform convex function with respect to D, if
J(;f;z;) = (1)zf0(z;)
f(z;)+(1 +zf00(z;)
f0(z;))q(z;):
We denote this class by UM(q).
Remark 3.2.Geometric interpretation: f2UM(q) if and only if J(;f;z;)
take all values in the convex domain included in right half plane D.
Remark 3.3.If we takeD=
(see Remark 2.2) we obtain the class UM.
Remark 3.4.From the above denition it easily results that q1(z;)q2(z;)
impliesUM(q1)UM(q2).
Theorem 3.5. For all;02[0;1]with<0we haveUM0(q)UM(q).

SUBCLASSES OF UNIFORMLY CONVEX FUNCTIONS 5
Proof. Fromf2UM0(q) we have
J(0;f;z;) = (10)zf0(z;)
f(z;)+0(1 +zf00(z;)
f0(z;))q(z;); (3.1)
whereq(z;) is univalent in Uwithq(0;) = 1,q0(0;)>0, and maps the unit
discUinto the convex domain included in right half plane D.
With the notationzf0(z;)
f(z;)=p(z;), where
p(z;) = 1 +p1(z;) +


;(z2U;2U);
we obtain:
J(0;f;z;) =p(z;) +0zp0(z;)
p(z;):
From (3.1) we have
p(z;) +0zp0(z;)
p(z;)q(z;)
withp(0;) =q(0;), Req(z;)>0, (z2U;2U).
In these conditions from Theorem 2.5, with = 0, we obtain p(z;)q(z;),
orp(z;) takes all values in D.
If we consider the function g: [0;0]U!C,
g(u;) =p(z;) +uzp0(z;)
p(z;);
withg(0;) =p(z;)2Dandg(0;) =J(0;f;z;)2D, since the geometric
image ofg(;) is on the segment obtained by the union of the geometric image
ofg(0;) andg(0;), we haveg(;)2Dorp(z;) +zp0(z;)
p(z;)2D.
ThusJ(;f;z;) takes all values in D, orJ(;f;z;)q(z;). This means
f2UM(q). 
Theorem 3.6. IfF(z;)2UM(q)thenf(z;) =LaF(z;)2S0(q), where
Lais the integral operator dened by (1.1) and2[0;1].
Proof. From (1.1) we have
(1 +a)F(z;) =af(z;) +zf0(z;);(z2U;2U):
With the notationzf0(z;)
f(z;)=p(z;), where
p(z;) = 1 +p1(z;) +


;(z2U;2U);
we have:
zF0(z;)
F(z;)=p(z;) +zp0(z;)
p(z;) +a:

6 R. S ENDRUT  IU
If we denotezF0(z;)
F(z;)=h(z;), withh(0;) = 1, we have from F(z;)2
UM(q) (see Denition 3.1)that:
h(z;) +zh0(z;)
h(z;)q(z;);
whereq(z;) is univalent in Uwithq(0;) = 1,q0(z;)>0, and maps the unit
discUinto the convex domain included in right half plane D.
From Theorem 2.5 we obtain h(z;)q(z;) orp(z;)+zp0(z;)
p(z;) +aq(z;).
Using the hypothesis and the construction of the function q(z;) we obtain from
Theorem 2.5 thatzf0(z;)
f(z;)=p(z;)q(z;) orf(z;)2S0(q)S.
The next denition (given in [1]) is adapted to the class H[a;n].
Denition 3.7. [1] Letf(z;)2Anand2[0;1],n2N. We say that fis an
nuniformly convex function with respect to D, if
Jn(;f;z;) = (1)In+1f(z;)
Inf(z;)+In+2f(z;)
In+1f(z;)q(z;):
We denote this class by UDn;(q).
Remark 3.8.Geometric interpretation: f2UDn;(q) if and only if Jn(;f;z;)
takes all values in the convex domain included in right half plane D.
Remark 3.9.If we consider D=D;
(see Remark 2.4) we obtain the class
UDn;(;
).
Remark 3.10.From the above denition it easily results that q1(z;)q2(z;)
impliesUDn;(q1)UDn;(q2).
Theorem 3.11. For all;02[0;1]with<0we haveUDn;0(q)UDn;(q).
Proof. Fromf2UDn;0(q) we have
Jn(0;f;z;) = (10)In+1f(z;)
Inf(z;)+0In+2f(z;)
In+1f(z;)q(z;); (3.2)
whereq(z;) is univalent in Uwithq(0;) = 1,q0(0;)>0, and maps the unit
discUinto the convex domain included in right half plane D.
With the notationIn+1f(z;)
Inf(z;)=p(z;), where
p(z;) = 1 +p1(z;) +


;(z2U;2U);

SUBCLASSES OF UNIFORMLY CONVEX FUNCTIONS 7
we have:
Jn(0;f;z;) =p(z;) +0zp0(z;)
p(z;):
From (3.2) we obtain
p(z;) +0zp0(z;)
p(z;)q(z;)
withp(0;) =q(0;), Req(z;)>0,z2U;2U.
In these conditions from Theorem 2.5 we obtain p(z;)q(z;), orp(z;)
takes all values in D.
If we consider the function g: [0;0]U!CU,
g(u;) =p(z;) +uzp0(z;)
p(z;);
withg(0;) =p(z;)2Dandg(0;) =Jn(0;f;z;)2D, it is easy to see that
g(;) =p(z;) +zp0(z;)
p(z;)2D:
Thus we have Jn(;f;z;)q(z;) orf2UDn;(q). 
Theorem 3.12. IfF(z;)2UDn;(q)thenf(z;) =LaF(z;)2S
n(q), where
Lais the integral operator dened by (1.1) .
Proof. From (1.1) we have
(1 +a)F(z;) =af(z;) +zf0(z;);(z2U;2U):
By means of the application of the linear operator Inwe obtain:
(1 +a)In+1F(z;) =aIn+1f(z;) +zIn+1f0(z;)
or
(1 +a)In+1F(z;) =aIn+1f(z;) +In+2f(z;):
With the notationIn+1f(z;)
Inf(z;)=p(z;), where
p(z;) = 1 +p1(z;) +


;(z2U;2U);
we have:
In+1F(z;)
InF(z;)=p(z;) +zp0(z;)
p(z;) +a:
If we denoteIn+1F(z;)
InF(z;)=h(z;), withh(0;) = 1, we have from F(z;)2
UDn;(q) (see Denition 3.7)that:
h(z;) +zh0(z;)
h(z;)q(z;);
whereq(z;) is univalent in Uwithq(0;) = 1,q0(0;)>0, and maps the unit
discUinto the convex domain included in right half plane D.

8 R. S ENDRUT  IU
From Theorem 2.5 we obtain h(z;)q(z;) orp(z;)+zp0(z;)
p(z;) +aq(z;).
Using the hypothesis we obtain from Theorem 2.5 that p(z;)q(z;) or
f(z;)2Sn(q). 
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1Faculty of Environmental Protection, University of Oradea, Str. B-dul
Gen. Magheru, No.26, 410048 Oradea, Romania
E-mail address :roxana.sendrutiu@gmail.com