R E S E A R C H
Open Access
On some differential inequalities in the unit
disk with applications
Adel A Attiya
**Correspondence:
[email protected] Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, 35516, Egypt
Abstract
In this paper we obtain a number of interesting relations associated with some differential inequalities in the open unit disk,U={z:|z|< 1}. Some applications of the main results are also obtained.
MSC: Primary 30C45; 30C80
Keywords: analytic functions; starlike functions; convex functions; spiral-like functions; Carathéodory functions
1 Introduction
LetAdenote the class of functions of the form
f(z) =z+
∞
n=
anzn (.)
which are analytic in the unit discU={z:|z|< }. Also, we denote byKthe class of func-tionsf(z)∈Athat are convex inU.
A functionf(z) in the classAis said to be in the classS∗(α) of starlike functions of order
α(≤α< ) if it satisfies
Re
zf(z)
f(z)
>α (z∈U) (.)
for someα(≤α< ). Also, we writeS() =S∗, the class of starlike functions inU. A functionf(z)∈Ais inSλ(|λ|<π
), the class ofλ-spiral-like functions, if it satisfies
Re
eiλzf
(z)
f(z)
> (z∈U). (.)
Definition . Letf(z) andF(z) be analytic functions. The functionf(z) is said to be sub-ordinate to F(z), writtenf(z)≺F(z), if there exists a functionw(z) analytic inU, with
w() = and|w(z)| ≤, and such thatf(z) =F(w(z)). IfF(z) is univalent, thenf(z)≺F(z)
if and only if f() =F() andf(U)⊂F(U).
LetDbe the set of analytic functionsq(z) injective onU\E(q), where
E(q) =ζ∈∂U:lim
z→ζq(z) =∞
andq(ζ)= forζ∈∂U\E(q). Further, letDa={q(z)∈D:q() =a}.
In this paper we obtain some interesting relations associated with some differential in-equalities inU. These relations extend and generalize the Carathéodory functions inU which have been studied by many authorse.g., see [–].
2 Main results
To prove our results, we need the following lemma due to Miller and Mocanu [, p.].
Lemma . Let q(z)∈Daand let p(z) =b+bnzn+· · ·
be analytic inUwith p(z)=b.If p(z)⊀q(z),then there exist points z∈Uandζ∈∂U\E(q)
and on m≥n≥for which
(i) p(z) =q(ζ), (ii) zp(z) =mζq(ζ).
Theorem . Let
P:U→C
with
ReaP(z)> (a∈C).
If p is a function analytic inUwith p() = and
Rep(z) +P(z)zp(z)> E
|a|Re(aP(z)), (.)
then
Reap(z)>α,
where
E= –Re(a) –αReaP(z)
+ ReaP(z) Im(a)+ αRe(a)
+Re(a) –αIm(a), (.)
withRe(a) >α.
Proof Let us defineq(z) andh(z) as follows:
q(z) =ap(z) and
h(z) = a– (α–a)z –z
The functionsqandhare analytic inUwithq() =h() =a∈Cwith
h(U) =w:Re(w) >α.
Now, we suppose thatq(z)⊀h(z). Therefore, by using Lemma ., there exist points
z∈U and ζ∈∂U\{}
such thatq(z) =h(ζ) andzq(z) =mζh(ζ),m≥n≥.
We note that
ζ=h–
q(z)
= q(z) –a
q(z) – (α–a)
(.)
and
ζh(ζ) =
–|q(z) –a|
Re(a–q(z))
. (.)
We haveh(ζ) =α+ρi(α,ρ∈R), therefore
Rep(z) +P(z)zp(z)
=Re
ah(ζ) +
aP(z)mζh
(ζ )
=Re
α+ρi
a
–m|α+ρi–a|
Re(a–α) Re
P(z)
a
≤Re
α+ρi
a
–|α+ρi–a|
Re(a–α) Re
P(z)
a
=Aρ+Bρ+C
=g(ρ), (.)
where
A= – Re(aP(z)) |a|Re(a–α),
B=Im(a)
|a|
+Re(aP(z))
Re(a) –α
and
C=
|a|
αRe(a) –α
+|a|– αRe(a)Re(aP(z ))
(Re(a) –α)
.
We can see that the functiong(ρ) in (.) takes the maximum value atρgiven by
ρ=Im(a)
+ Re(a) –α
Re(aP(z))
Hence, we have
Rep(z) +P(z)zp(z)
≤g(ρ)
= E |a|Re(aP(z)),
whereEis defined by (.). This is in contradiction to (.). Then we obtainRe(ap(z)) >α.
Theorem . Let p(z)a nonzero analytic function inUwith p() = .If
p(z) +zp
(z)
p(z) –
<Re(a–α)
|a| p(z), (.) then
Re
a p(z)
>α,
whereRe(a) >α.
Proof Let us define bothq(z) andh(z) as follows:
q(z) =a/p(z) and
h(z) =a– (α–a)z –z
Re(a) >α.
The functionsqandhare analytic inUwithq() =h() =a∈Cwith
h(U) =w:Re(w) >α.
Now, we suppose thatq(z)⊀h(z). Therefore, by using Lemma ., there exist points
z∈U and ζ∈∂U\{}
such thatq(z) =h(ζ) andzq(z) =mζh(ζ),m≥n≥.
We note that
ζh(ζ) =
–|q(z) –a|
Re(a–q(z))
. (.)
We haveh(ζ) =α+ρi(ρ∈R); therefore,
|p(z) +zp
(z) p(z) – | |p(z)|
=α+ρi
a – m
a
|a–α–iρ|
Re(a–α) –
≥|
a|
≥ |a|
|a–α–iρ|
Re(a–α) +Re(a–α)
≥
|a|Re(a–α)
Re(a–α)+Im(a) –ρ ≥Re(a–α)
|a| .
This is in contradiction to (.). Then we obtainRe(pa(z)) >α.
3 Applications and examples
PuttingP(z) =β(β> ; real) in Theorem . we have the following corollary.
Corollary . If p is a function analytic inUwith p() = and
Rep(z) +βzp(z)> E β|a|Re(a),
then
Reap(z)>α,
where
E= –Re(a) –αβRe(a)+ βRe(a) Im(a)+ αRe(a)+Re(a) –αIm(a),
withRe(a) >α(α≥).
Puttingβ= in Corollary ., we obtain the following corollary.
Corollary . If p is a function analytic inUwith p() = and
Rep(z) +zp(z)> Re(a)
Re(a) –α–Re(a)
|a|
Re(a) – α,
then
Reap(z)>α,
withRe(a) >α(α≥).
Corollary . Let f(z)∈A, (g(z))a∈S∗and
Re
f(z)
g(z)
> E |a|Re(ag(z)
zg(z))
,
then
Re
af(z) g(z)
>α,
Proof Puttingp(z) = fg((zz)) andP(z) =zgg(z(z))in Theorem ., we have
Rep(z) +P(z)zp(z)=Re
f(z)
g(z)
.
Since (g(z))a∈S∗, which givesRe(azg(z)
g(z) ) > , therefore,Re(aP(z)) > . This completes the
proof of the corollary.
Example . Letf(z)∈Aand
Ref(z)> Re(a)
Re(a) –α–Re(a)
|a|
Re(a) – α,
then
Re
af(z) z
>α,
whereRe(a) >α.
Example . Letf(z)∈Aand
Re
+zf
(z)
f(z) –
zf(z)
f(z)
zf(z)
f(z)
> Re(a)
Re(a) –α–Re(a)
|a|
Re(a) – α,
then
Re
azf
(z)
f(z)
>α,
whereRe(a) >α.
() Puttinga=eiλ(|λ|<π
) andα= in Theorem ., we have Theorem due to Kim and Cho [].
() Puttinga=eiλ(|λ|<π
),P(z) =β(β> ; real) andα= in Theorem ., we have Corollary due to Kim and Cho [].
() Puttinga=α= andP(z) = in Theorem ., we have the result due to Nunokawa
et al.[].
() Puttinga=eiλ(|λ|<π
),P(z) = andα= in Theorem ., we have Corollary due to Kim and Cho [].
Puttingp(z) =zff((zz)) in Theorem ., we have the following corollary.
Corollary . Let p(z)a nonzero analytic function in U with p() = .If
zff((zz))
<Re(a–α) |a|
zff((zz)),
then
Re
a zf(z)
f(z)
>α,
Remark
() Puttinga= andα= in Corollary ., we have the result due to Attiya and Nasr [].
() Puttinga= andα= in Corollary ., we have the result due to Kim and Cho [].
Competing interests
The author declares that he has no competing interests.
Acknowledgement
The author would like to express his gratitude to the referee(s) for the valuable advices to improve this paper.
Received: 13 December 2013 Accepted: 6 January 2014 Published:24 Jan 2014
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