ALIAGA, EDMOND

BULBOACA, TEODOR

PETRUSHEVSKI, MIRKO

TUNESKI, NIKOLA

English

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.1, 713
ISSN 1857-8365, UDC: 517.546/.547
SOME RESULTS OVER THE FIRST DERIVATIVE OF ANALYTIC
FUNCTIONS
NIKOLA TUNESKI, TEODOR BULBOAC, MIRKO PETRUEVSKI AND EDMOND ALIAGA
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. Let f be an analytic function in the open unit disk and normalized such
that f(z) = z+ anzn + : : : ; n 2 N; n  2: In this work we use dierential subordina-
tions to study the expression
z
f 0(z)  1
f(z)  z
and give estimates of jf 0(z)  1j: Also, sucient conditions for a function to be with
bounded turning are obtained and some open problems are posed. This work is a
continuation of the results published in [7].
1. Introduction and preliminaries
Let H(D) be the class of functions that are just analytic in the unit disk D = fz 2
C : jzj < 1g and let An, n 2 N, n  2; be its subclass consisting of functions f that
are analytic in D and normalized such that f(z) = z + anz
n + : : : ; an 6= 0: Further,
f 2 A  A1 is such that f(0) = f 0(0)  1 = 0:
Now, f 2 A is a starlike function if and only if Re[zf 0(z)=f(z)] > 0; z 2 D: All such
functions are univalent and the corresponding class is denoted by S: Another subclasses
of univalent functions are R (0   < 1) and R() (0 <   1) consisting of functions
f 2 A such that Re f 0(z) >  (z 2 D) and jarg f 0(z)j < =2 (z 2 D); respectively. The
special case R  R0 = R(1) is the well known class of functions with bounded turning.
The interest for this class comes from the result of Krzyz [6] that S does not contain R
and R does not contain S: In this paper, using a method from the theory of dierential
subordinations (valuable references on this topic are [1] and [2]), we will receive criteria
over the expression
z
f 0(z)  1
f(z)  z
2010 Mathematics Subject Classication. 30C45, 30C50.
Key words and phrases. Analytic function, bounded turning function, univalent function, dierential
subordination.
7
8 N. TUNESKI, T. BULBOAC, M. PETRUEVSKI AND E. ALIAGA
that will embed a function f 2 An, n 2 N, n  2, in the class R: This work is a
continuation of the results published in [7]. From the theory of rst-order dierential
subordinations we will use the following lemma:
Lemma 1.1. [4] Let q be univalent in the unit disk D, and let (w) and (w) be
analytic in a domain D containing q(D), with (w) 6= 0 when w 2 q(D). Set Q(z) =
zq0(z)(q(z)), h(z) = (q(z)) +Q(z), and suppose that:
(i) Q is starlike in the unit disk D,
(ii) Re
zh0(z)
Q(z)
= Re

0(q(z))
(q(z))
+
zQ0(z)
Q(z)

> 0, z 2 D.
If p is analytic in D, with p(0) = q(0), p(D)  D and
(1.1) (p(z)) + zp0(z)(p(z))  (q(z)) + zq0(z)(q(z)) = h(z)
then p(z)  q(z), and q is the best dominant of (1.1)).
2. Main results and consequences
Using Lemma 1.1 we will receive conclusions that will later lead to criteria for a function
f to be in the class R.
Theorem 2.1. Let f 2 An, n 2 N, n  2, such that f(z) 6= z for all z 2 D n f0g, and
let an =
f(n)(0)
n! . If 0 < jj  janj and
(2.1) z
f 0(z)  1
f(z)  z   n 
z
an + z
=: h1(z);
then
(2.2)
f(z)  z
zn
 an + z;
and the function an + z is the best dominant of (2.1). Even more,
(2.3)
f(z)  zzn   an
 < ; z 2 D;
and this conclusion is sharp, i.e. in the inequality (2.3) the parameter jj can not
be replaced by a smaller number so that the implication holds.
Proof. Let choose (w) = 0; (w) = 1w ; p(z) =
f(z) z
zn and q(z) = an + z: Then
;  2 H(D), where D = C := C n f0g; and D  q(D) since 0 < jj  janj: Further,
Q(z) = h(z) = zq0(z)(q(z)) =
zq0(z)
q(z)
=
z
an + z
and
Re
zh0(z)
Q(z)
= Re
zQ0(z)
Q(z)
= Re
1
1 + =anz
> 0; z 2 D:
So, q is univalent in D; p 2 H(D), p(0) = q(0) = an and p(z) 6= 0 for all z 2 D,
i.e. p(D)  D, and all the conditions of Lemma 1.1 are satised. Concerning that
the subordinations (1.1) and (2.1) are equivalent, we receive subordination (2.2) and its
equivalent inequality (2.3).
SOME RESULTS OVER THE FIRST DERIVATIVE OF ANALYTIC FUNCTIONS 9
For the sharpness of our result, let assume that subordination (2.1) and inequality f(z) zzn   an < j1j, z 2 D, hold, i.e. f(z) zzn  an + 1z. But, the function an + z is
the best dominant of (2.1), meaning that an + z  an + 1z, i.e. jj  j1j. 
It is easy to verify that if 0 < jj < janj, then h1(D) (where h1 was dened in (2.1)) is
an open disk with the center
(2.4) c =
1
2

h
h1

ei arg(an=)

+ h

 ei arg(an=)
i
=
jj2
jj2   janj2 ;
and radius
(2.5) r =
h1 ei arg(an=)  c = jj  janjjanj2   jj2 :
Therefore, Theorem 2.1 brings the following corollary.
Corollary 2.1. Let f 2 An, n 2 N, n  2, such that f(z) 6= z for all z 2 D n f0g, and
let an =
f(n)(0)
n! . If 0 < jj < janj andz f 0(z)  1f(z)  z   n  jj
2
jj2   janj2
 < jj  janjjanj2   jj2 ; z 2 D;
then f(z)  zzn   an
 < jj; z 2 D:
This implication is sharp (the radius of the open disk from the conclusion is the
smallest possible so that the corresponding implication holds) due to the function
f(z) = z + anz
n + zn+1; an 6= 0:
In the case when n = 2 we receive
Corollary 2.2. Let f 2 A2, and  2 C with 45 ja2j  jj < ja2j, where a2 = f
00(0)
2 .
Also, let denote
 :=
8<
:
 2 + jjja2j jj ; if 45 ja2j  jj 
q
2
3 ja2j
2 + jjja2j+jj ; if
q
2
3 ja2j  jj < ja2j
:
If
(2.6) jf 0(z)  1j < 
f(z)z   1
 ; z 2 D n f0g;
then
(2.7)
f(z)  zz2   a2
 < 1 := jj; z 2 D;
and
(2.8) jf 0(z)  1j < 2; z 2 D;
where
2 :=
8<
:
(ja2j+jj)(2ja2j 3jj)
jj ja2j ; if
4
5 ja2j  jj 
q
2
3 ja2j;
2ja2j+ 3jj; if
q
2
3 ja2j  jj < ja2j:
10 N. TUNESKI, T. BULBOAC, M. PETRUEVSKI AND E. ALIAGA
Moreover, the implication (2.6) ) (2.7) is sharp for
q
2
3 ja2j  jj < ja2j, and the
implication (2.6) ) (2.8) is sharp for
q
2
3 ja2j  jj < ja2j, i.e. for these ranges of
jj, the values 1 and 2 are the smallest ones so that the corresponding implications
hold.
Also, if 2 < 1, then f is univalent with bounded turning, i.e. f 2 R1 and
f 2 R(2), where 1 = 1  2 and 2 = arcsin 2.
Proof. First we will prove inequality (2.7). The assumption (2.6) leads to
jf 0(z)  1j <  
f(z)z   1
 =  
f(z)  zz
 ; z 2 D n f0g;
meaning that f(z) zz 6= 0 for all z 2 D n f0g, hence f(z) 6= z for all z 2 D n f0g. Also, the
inequality (2.6) implies z f 0(z)  1f(z)  z
 < ; z 2 D n f0g;
and letting z ! 0 in the above inequality we obtain that   2 is a necessary condition
for the above inequality to hold in the case z = 0.
It is easy to check that
 =
jj  ja2j  
3jj2   2ja2j2
ja2j2   jj2 = r   j2 + cj;
where c and r are dened as in (2.4) and (2.5), respectively, and that   2 whenever
jj  45 ja2j. Further, we can writez f 0(z)  1f(z)  z   (2 + c) + (2 + c)
 < ; z 2 D n f0g;
and it follows thatz f 0(z)  1f(z)  z   (2 + c)
 < + j2 + cj = r; z 2 D n f0g:
The above inequality holds for z = 0, since jcj <  + j2 + cj = r for 0 < jj < ja2j, and
thus, from the rst part of the Theorem 2.1'(i) for the special case n = 2 we get (2.7)).
From the assumption (2.6) we get
(2.9) jf 0(z)  1j <  
f(z)z   1
 <  
f(z)  zz2
 ; z 2 D n f0g;
and the inequality (2.8) follows from (2.7) and (2.9)), having in mind that 2 =  (ja2j+).
The implication (2.6) ) (2.7) is sharp for
q
2
3 ja2j  jj < ja2j, and the implication
(2.6)) (2.8) is sharp for
q
2
3 ja2j  jj < ja2j, since for the function f(z) = z+a2z2+z3
we have:
jf 0(z)  1j = jzj  j2a2 + 3zj < 2ja2j+ 3jj; z 2 D;f(z)z   1
 = jzj  ja2 + zj
and f(z)  zz2   a2
 = jj  jzj < jj; z 2 D:
SOME RESULTS OVER THE FIRST DERIVATIVE OF ANALYTIC FUNCTIONS 11
The assertion (2.6) is equivalent to
 >
3z + 2a2z + a2
 ; z 2 D n f0g;
and a simple computation shows that
sup
3z + 2a2z + a2
 : z 2 D n f0g

= 2 +
jj
ja2j+ jj ;
whenever jj < ja2j, hence
  2 + jjja2j+ jj ;
which holds for
q
2
3 ja2j  jj < ja2j.
Since
 2 + jjja2j   jj < 2 +
jj
ja2j+ jj ; if jj <
r
2
3
ja2j;
the function f(z) = z+ a2z
2 + z3 shows that the implication (2.6) ) (2.7) is not sharp
for 45 ja2j < jj <
q
2
3 ja2j.
Finally, from (2.7) and the denitions of the classes R and R() we receive f 2 R1
and f 2 R(2). 
For 2 = 1; Corollary 2.2 reduces to the next example:
Example 2.1. Let f 2 A2, with 15 < ja2j  518 , where a2 = f
00(0)
2 . Also, let
 :=
(
3
1+ja2j ; if 0:2 =
1
5 < ja2j  12+p6 = 0:22474 : : : ;
1
jj+ja2j ; if
1
2+
p
6
 ja2j  518 = 0:27 : : : ;
where
jj :=  (1 + ja2j) +
p
25ja2j2 + 14ja2j+ 1
6
:
If
(2.10) jf 0(z)  1j <  
f(z)z   1
 ; z 2 D n f0g;
then
(2.11) jf 0(z)  1j < 1; z 2 D:
This implication is sharp for 15 < ja2j  12+p6 = 0:22474 : : :. Also, the function f is
univalent with bounded turning, i.e. f 2 R.
Proof. We need to prove that conditions of Corollary 2.2, in the case 2 = 1, are equiva-
lent to the assumptions of this example.
For the case when 45 ja2j  jj 
q
2
3 ja2j, then 2 =  (ja2j+ jj) = 1 if and only if
 = 1jj+ja2j , i.e.
 2 (jj+ ja2j) +  (ja2j+ jj)ja2j   jj = 1;
or in other words
jj = jj :=   (1 + ja2j) +
p
25ja2j2 + 14ja2j+ 1
6
:
12 N. TUNESKI, T. BULBOAC, M. PETRUEVSKI AND E. ALIAGA
Here, we considered only the positive sign of the square root since the negative one leads
to negative values of jj. Further, the inequalities
4
5
ja2j  jj = jj 
r
2
3
ja2j
are equivalent to
0:22474 : : : =
1
2 +
p
6
 ja2j  5
18
= 0:27 : : : :
In a similar way, for the case
q
2
3 ja2j  jj < ja2j we have 2 = 1 if and only if
3jj+ 2ja2j = 1, i.e. jj = 1 2ja2j3 . A simple calculus shows thatr
2
3
ja2j  jj < ja2j
is equivalent to
0:2 =
1
5
< ja2j  1
2 +
p
6
= 0:22474 : : : ;
which completes the proof. 
Remark 2.1. Weather implications (2.6)) (2.8) for 45 ja2j  jj <
q
2
3 ja2j (Corollary
2.2 and (2.10) ) (2.11) for 15 < ja2j  12+p6 = 0:22474 : : : (Example 2.1 are sharp
are still open problems.
References
[1] T. Bulboac : Dierential subordinations and superordinations. New results, House of Science
Book Publ., Cluj-Napoca, 2005.
[2] S.S Miller, P.T.Mocanu: Dierential subordinations, Theory and Applications,Marcel Dekker,
New York-Basel, 2000.
[3] S.S Miller, P.T.Mocanu: Dierential subordinations and univalent functions, Michigan Math.
J. 28 (1981), 157-171.
[4] S.S Miller, P.T.Mocanu: On some classes of rst-order dierential subordinations, Michigan
Math. J., 32 (1985), 185-195.
[5] R.W. Ibrahim, M. Darus: Extremal bounds for functions of bounded turning, Int. Math. Forum,
6(33) (2011), 1623-1630.
[6] J.Krzyz: A counter example concerning univalent functions, Folia Societatis Scientiarum Lubli-
nensis, 2 (1962), 57-58.
[7] N.Tuneski, M. Obradovi¢: Some properties of certain expression of analyti functions, Comput.
Math. Appl., 62 (2011), 3438-3445.
Faculty of Mechanical Engineering
Ss. Cyril and Methodius University in Skopje
Karpo² II b.b.,1000 Skopje, Republic of Macedonia
E-mail address: nikola.tuneski@mf.edu.mk
Faculty of Mathematics and Computer Science
Babe³-Bolyai University
400084 Cluj-Napoca, Romania
E-mail address: bulboaca@math.ubbcluj.ro
Faculty of Mechanical Engineering
Ss. Cyril and Methodius University in Skopje
Karpo² II b.b., 1000 Skopje, Republic of Macedonia.
E-mail address: Mirko.Petrushevski@mf.edu.mk
SOME RESULTS OVER THE FIRST DERIVATIVE OF ANALYTIC FUNCTIONS 13
Faculty of Mathematics and Natural Sciences
University of Prishtina
Nëna Terezë n.n., 10000 Prishtina, Prishtina, Kosovo
E-mail address: edmond.aliaga@uni-pr.edu


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