Let 1 A denote the class of functions      2 ( ) n n n f z z a z analytic in the unit disc E  {z : |z| <1}.  M denotes the class of functions in 1 A which satisfy the conditions 0 ( ). ( )   z f z f z and for 0  1, 0 ( ) ( ( )) ( ) ( ) ) 1 ( Re            f z zf z f z zf z   . We are interested in determining the sharp upper bound for the functional 2 2 4 3 a a  a for the class  M .

1. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 19|P a g e
Hankel Determinent for Certain Classes of Analytic Functions
Gurmeet Singh
KHALSA COLLEGE, PATIALA
ABSTRACT:
Let 1A denote the class of functions 



2
)(
n
n
n zazzf analytic in the unit disc :{zE  |z| <1}.
M denotes the class of functions in 1A which satisfy the conditions 0
)().(


z
zfzf
and for
10  , 0
)(
))((
)(
)(
)1(Re 










zf
zfz
zf
zfz
 . We are interested in determining the sharp upper bound
for the functional
2
342 aaa  for the class M .
MATHEMATICS SUBJECT CLASSIFICATION: 30C45.
KEYWORDS: Analytic functions, univalent functions, starlike functions, convex functions,  convex
functions, Bazilevic functions.
I. INTRODUCTION:
Let 1A be the class of functions




2
)(
n
n
n zazzf (1.1)
analytic in the unit disc :{zE  |z| <1 }.
S denotes the Class of functions




2
)(
n
n
n zazzf (1.2)
analytic and univalent in :{zE  |z| <1 }.
Let  p be the class of functions of the form
  ...1 3
3
2
21  zpzpzpz (1.3)
analytic in the unit disc :{zE  |z| < 1 } with Re   0zP . Carathéodory [1] introduced the class  p .
Noshiro [2] and Warschawiski [3] introduced the class of univalent functions
 fR { 1A : Ezzf  ,0)(Re } (1.4)
known as N-W class of functions.
R and its subclasses were studied by several authors including Goel and Mehrok [11, 12].
*
S { f 1A : Ez
zf
zfz


,0
)(
)(
Re } (1.5)
is the class of starlike univalent functions .
RESEARCH ARTICLE OPEN ACCESS

2. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 20|P a g e
K = { f 1A : Ez
zf
zfz



,0
)(
))((
Re } (1.6)
is the class of convex univalent functions.
Macanu [5] introduced the class of  - convex functions defined as
   
   
 
  
  































Ez
zf
zfz
zf
zfz
z
zfzf
Af
M
,10,01Re
,0:1


(1.7)
For any real , Miller, Macanu and Reade [7] have shown that all  - convex functions, are starlike in E; and
for all 1 , all  -convex functions are convex in E.
Hassoon, Al-Amiri and Reade [9] introduced the class of analytic functions
      
  


















 Ez
zf
zfz
zfAfH ,10,01Re:1  (1.8)
Bazilevic [4] introduced the following class of analytic univalent functions. For  real, 0 ,    pzP 
and   
Szg
       























tz
dttgttPiAfgPB
1
0
1
1 :,,, (1.9)
Taking 0 and   zzg  in (1.9), we get  zPB ,,0, as the class of functions
   
















 




1
0
1
1 :,,0,
z
dtzzPAfzPB (1.10)
The class  zPB ,,0, was studied by Singh [6] and El-Ashwah and Thomas [13].
   















Ez
z
zf
zfAfB ,10,0Re:
1
1 

 (1.11)
is also a subclass of Bazilevic functions.
II. Preliminary Lemmas.
Lemma 2.1[15]: Let    pzpzpzpz  ...1 3
3
2
21 , then
2np for all n (n=1,2,3...).
Lemma 2.2[9]: If    pzpzpzpz  ...1 3
3
2
21 , then
 xppp 2
1
2
12 42  and
      zxpxppxpppp
22
1
22
11
2
11
3
13 1424424 
for some x and z with 1,1  zx .
III. Main Results.
Theorem 3.1: Let Mf  , then

3. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 21|P a g e
 
      
;10,
21
1
7241522131
13
2322
3
2
342 




 


aaa (3.1)
And 0,12
342  aaa (3.2)
Results are sharp.
Proof: Since Mf  , it follows that
 zP
zf
zfz
zf
zfz






)(
))((
)(
)(
)1(  (3.3)
Equating the coefficients in (3.3), it is easily established that
 
 
 
  
 
 
   
 
    
























3
4
1
2
213
4
2
2
12
3
1
2
131216
1761
312112
51
313
1212
31
312
1









pppp
a
pp
a
p
a
(3.4)
System (3.4) yields
 
           
          22
12
24
1
2
2
2
1
2
31
32
2
342
312213131761218
21512112412141
ppp
pppp
C
aaa


 

 (3.5)
 
    42
1213148
1



C . (3.6)
Using lemma 2.2 in (3.5), we get
 
           
      
          22
1
22
1
24
1
2
2
1
2
1
2
1
2
22
1
22
11
2
11
3
11
32
2
342
41833131761218
41512112
1424421214
1
xppp
xppp
zxpxppxpppp
C
aaa








(3.7)
Replacing 1p by  2,0p , (3.7) takes the form
 
       
     
       
   
 
         
      zxpp
xp
xpp
p
C
aaa
2232
2223
22
22
232
4
222
232
2
342
141218
7413111241
4
183316
15121121218
833131761218
15121121214
1

































(3.8)
Applying triangular inequality to (3.8), we obtain

4. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 22|P a g e
 
       
     
       
   
 
          
       














































232
2223
22
22
232
4
222
232
2
342
41218
7413116241
4
183316
15121121218
833131761218
15121121214
1
pp
xpp
xpp
p
C
aaa







(3.9)
 
  1,
1
 xF
C


. (3.10)
  0 F and therefore  F is increasing in [0,1] and  F attains its maximum value at
1 x .
Putting 1x in (3.9) , we have
 
       
     
       
   
 
         
















































2223
22
22
232
4
222
232
2
342
7413111241
4
183316
15121121218
833131761218
15121121214
1
pp
pp
p
C
aaa






 
          
          
       
          
   
  
























































4
2
23
42322
232
4
2322
322222
13148
73114
1311274114183316
15121121218
4
741117612181512124
1211218331683313
1







p
p
C
 
       
 
  







pG
C
pBpA
C



1
13148
1 424
(3.11)
where     21524714 23
 A , (3.12)
and     .148
4
 B (3.13)
      024 3
 pBpApG  (3.14)
which implies that p=0 or
 
 

A
B
p
2
2
 .

5. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 23|P a g e
p=0 does not give maximum value and is rejected.
 
 
 
 32
3
2
724152
16
2 






A
B
p , )0(  . (3.15)
gives the maximum value of  pG .
Putting the value of
2
p from (3.15) in (3.11) , we get
 
 
 
   






4
2
2
342 13148
4
1



 A
B
C
aaa , 0 (3.16)
Substituting the values from (3.6), (3.12) and (3.13) in (3.16) , the bound (3.1) follows.
Consider the case 0 . In this case,     0,0   BA and   48C .
Putting these values in (3.11) , we get 12
342  aaa .
Result (3.1) is sharp for
 
 32
3
1
724152
16




p , 12 p and 3p obtained from (3.5).
Result (3.2) is sharp for 1,0 21  pp and 23 p .
Remark 3.1: Taking 1 in (3.1) , we get
8
12
342  aaa , a result due to Janteng etal.[15].
Remark 3.2: Result (3.2) is also due to Janteng etal.[15].
Theorem 3.2: Let Hf  , then
  17
5
0,
1
1
9
4
2
2
342 

 

aaa ; (3.17)
and
 
    
1
17
5
,
19
4
,
4720121144
517
232
2
2
342 




 


aaa . (3.18)
Results are sharp.
Proof: Since Hf  , therefore by definition       
 
 zP
zf
zfz
zf 


 1 .
Identification of terms in the above equation yields
 
 
    
 
  



























2114
12
2114
3
214
13
2
3
1213
4
2
12
3
1
2
pppp
a
pp
a
p
a
(3.19)
From (3.19), we obtain
 
       
     22
12
4
1
2
2
131
2
2
342
2221811236
21544191
ppp
pppp
C
aaa


 

 , (3.20)

6. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 24|P a g e
Where  
  2
121288
1



C . (3.21)
By lemma 2.2, we get
 
         
          
   4
1
22
1
2
1
2
1
2
1
2
1
22
1
22
11
2
11
3
11
2
2
342
11236
4212184154
14244219
1
p
xppxppp
zxpxppxpppp
C
aaa







 (3.22)
1x and 1z . Changing 1p to  2,0p , (3.22) takes the form
 
     
      
    zxpp
pp
xppp
C
aaa
222
222
222432
2
342
14118
92121324
44131283121
1








 
     
       
    













22
2222
222432
4118
921211624
44131283121
1
pp
ppp
ppp
C



  
 

F
C
1
 , 1 x . (3.23)
  0 F and therefore  F is increasing in [0,1] and maximum  F =  1F .
Putting the value 1 in (3.23), we arrive at
 
     
       









 222
222432
2
342
92121324
441312831211
pp
ppp
C
aaa



 
      
        












 21128921421321318
92141312831211
222
42232
p
p
C
 
      
 
  







pG
C
pBpA
C



1
21128
1 24
(3.24)
Where     0972012 32
 A in [0, 1] (3.25)
And     51714  B . (3.26)
Case I:
17
5
0  so that   0B .
  0 pG and  pG attains its maximum value at 0p .
From (3.21) and (3.24) it follows that
 2
2
342
1
1
9
4

 aaa .
Result is sharp for 1,0 21  pp and 23 p .

7. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 25|P a g e
Case II: 1
17
5
 so that   0B .
      024 3
 pBpApG  which implies that
0p or
 
 
  
 32
2
47201
5171
2 






A
B
p (3.27)
0p does not give the maximum value and is rejected.
Substituting the value of
2
p from (3.27) in (3.24), we conclude that
 
 
 
 





 



21128
4
1 2
2
342
A
B
C
aaa (3.28)
Putting the values from (3.21), (3.25) and (3.26) in (3.28), result (3.18) follows.
Equality sign in (3.18) holds for
  
  1,
47201
5171
2321 


 pp


and 3p obtained from (3.20).
Remark 3.3: Letting 0 in (3.17) , we get
9
42
342  aaa , a result proved by Janteng etal.[7] for the class R.
Remark 3.4: Letting 1 in (3.18), it follows that
8
12
342  aaa , result proved by Janteng etal.[8] for the class K
Theorem 3.3: Let Bf  , then
 2
2
342
2
4



aaa . (3.29)
The result is sharp.
Proof: Since Bf  , therefore it follows that
       10,
1









zP
z
zf
zf . (3.30)
Equating the coefficients in (3.30) , we get
 
 
 
 
 
 
  
  
  
























3
3
1213
4
2
2
12
3
1
2
16
121
21
1
3
12
1
2
1









pppp
a
pp
a
p
a
(3.31)
(3.31) gives
 
           
            22
12
4
1
2
21
2
31
32
2
342
211233231212
2121641231
ppp
pppp
C
aaa






(3.32)
Where  
    42
12312
1



C . (3.33)

8. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 26|P a g e
Using lemma 2.2 in (3.32), we obtain
 
           
       
            22
1
22
1
4
1
2
2
1
2
1
2
1
2
22
1
22
11
2
11
3
11
32
2
342
41333231212
413216
142442123
1
xppp
xppp
zxpxppxpppp
C
aaa








(3.34)
Changing 1p to  2,0p , (3.34) takes the form
 
        
      
        
   
 
       
      zxpp
xpp
xpp
p
C
aaa
2232
2223
22
22
232
4
32
232
2
342
14126
314413
4
316
13216126
33231212
13216123
1
































(3.35)
Coefficient of
4
p in (3.35) changes from negative to positive in [0, 1] and therefore
there must exist 0 in (0, 1) so that coefficient of
4
p is negative for 00   and positive for
10   .
Case I: 00   , from (3.35) it follows that
 
          
    
        
   
 
        
      














































232
223
22
22
232
4
2
3223
2
342
4126
3122413
4
31
132112
6
231212
1231321633
1
pp
xppp
xpp
p
C
aaa







 
  1,
1
 xF
C


. (3.36)
  0 F and therefore  F is increasing in [0,1] and
 F will attain its maximum value at 1 x .
Putting 1 x in (3.36), we get
 
          
          
        
         
 
  














































4
22
322
232
4
3232
2223
2
342
1348
4
1311312
13212122
12
13231212129
13211213633
1






pp
p
C
aaa

9. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 27|P a g e
 
       
 
  







pG
C
pBpA
C



1
13148
1 424
1
. (3.37)
Where
    
     





01412
0202161
3
23
1


B
A
. (3.38)
  0 pG ,  pG is decreasing in [0, 2] and maximum  pG =     4
131480  G .
From (3.37) and (3.33), it follows that
 2
2
342
2
4



aaa .
Case II: 10   , proceeding as in case I, from (3.35) , we get
 
         
        
     























42
4
3332
222
2
342
13148
13123
231212126
1




pB
p
C
aaa .
Where  B is given by (3.38))
 
       424
2 13148
1


 pBpA
C
. (3.39)
where     074133418 5432
2  A .
As discussed in case I,
 2
2
342
2
4



aaa .
Combining both the cases, proof of the theorem is complete.
Result (3.29) is sharp for 01 p , 12 p and 23 p .
Remark 3.5: Putting 0 in (3.29) , we have
12
342 aaa , a result established by Janteng etal.[15] for the class

S .
Remark 3.6: If we put 1 in (3.29), we get
9
42
342  aaa , a result established by Janteng etal.[14] for the class R.
References
[1] C. Carathéodory, Über Den Variabilitatsb Erecöch Der Fourieŕ Schen Konstanten Von Positivenen
harmonischen-Funktionan, Rendiconti Palerno, 32(1911), 193-217.
[2] K.Noshiro, On the theory of schlicht functions,J. Fac. Soc. Hokkaido univ.Ser.1,2(1934-1935), 129-155.
[3] S. Warchawski, On the higher derivative at the boundary in conformal mapping, Trasaction Amer.
Math.Soc.,38(1935), 310-340.
[4] I.E.Bazilevic, On the case of integrability in quadrature of the Lowner equation, Mat Sb. 37(1958), 471-
476.
[5] P.T.Mocanu , Une propritéde convexité generalise dans la théorie de la representation
conforme,Mathematica(CLUJ), 34 (1969), 127-133.
[6] R.Singh, On Bazilevic functions, Proceeding Amer. Math.Soc.,38(1973), 261-263.
[7] S.S.Miller ,P.T.Mocanu and M.O.Reade, All  -convex functions are univalent and starlike, Proceeding
Amer. Math.Soc.,37(1973), 553-554.
[8] Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Gottingen,1975.

10. Gurmeet Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 10( Part - 3), October 2014, pp.19-28
www.ijera.com 28|P a g e
[9] S.Hassoon,Al-Amiri and M.O.Reade, On A linear combination of some expressions in the theory of the
univalent functions, Monatshafte fur Math.,80(1975),257-264.
[10] R.J.Libera and E.J.Zlotkiewicz, Early coefficients of the inverse of a regular convex function,
Proceeding Amer. Math.Soc., 85(1982) , 225-230.
[11] R.M.Goel and B.S.Mehrok, A subclass of univalent functions, Houston J.Math.,8(1982),343-357.
[12] R. M. Goel, Beant Singh Mehrok, A subclass of univalent functions, J.Austral.Math.Soc. (Series
A),35(1983), 1-17.
[13] R.M.El-Ashwah and D.K.Thomas, Growth results for a subclass of Bazilevic functions, Inter J.Math.ana
Math Sci, 8(1985), 785-792.
[14] A.Janteng , S.A.Halim and M.Darus, Coefficient inequality for a function whoose derivative has a
positive real part, J.Ineq. in pure and appl. Math., 7(2006), 1-4.
[15] Aini Janteng, Hankel determinant for starlike and convex functions, Int. J.Math.Anal., 1(2007) , 619-
625.