Advances in Theoretical and Applied Mathematics
ISSN 0973-4554 Volume 9, Number 2 (2014), pp. 129-132
© Research India Publications
http://www.ripublication.com
On Strictly Concave Functions
Banyat Sroysang
Department of Mathematics and Statistics, Faculty of Science and Technology,
Thammasat University, Pathumthani 12121 Thailand
[email protected]
Abstract
Any strictly concave continuous real-valued function f on a closed interval I
f(x)+f(y)  x+y
must preserve the property
<f 
 for all x,y∈I. In this paper, we
2
 2 
give sufficient conditions for being a strictly concave function.
Mathematics Subject Classification: 52A40, 52A41
Keywords: strictly concave function, inequality
1 Introduction
Let f be a continuous real-valued function on a closed interval I. We say that f is
strictly concave if
f(x)+(1−t)f(y)<f(tx+(1−t)y)
for all x,y∈I and t∈ [0,1 ].
Moreover, f is strictly concave if and only if
f(x)+f(y)  x+y
<f 

2
 2 
for all x,y∈I.
In 2012, Sulaiman [Error! Reference source not found.] presented some
properties on strictly concave functions. In this paper, we give sufficient conditions
for being a strictly concave function.
130
Banyat Sroysang
2 Results
In this section, we denote the set of all continuous real-valued functions on a closed
interval I by CRF (I).
n
Theorem 2.1 Let f1,f2,...,fn∈CRF (I)
n
be such that  fi≥0 and
i=1
n
 fi(x)+  fi(y)
i=1
i=1
2
n  x+y
<mini=1,2,...n fi 

 2 
n
for all x,y∈I. Then  fi is strictly concave.
i=1
Proof. Let x,y∈I. Then
 x+y
 x+y
fj 
=mini=1,2,...n fi  2 
2




for some j∈ {1,2,...n
}.
We obtain that
n
n
 fi(x)+  fi(y)
i=1
i=1
n  x+y
<mini=1,2,...n fi 

 2 
n
  x+y
= fj 

  2 
n
 x+y
=  fj 

 2 
i=1
n
 x+y
≤  fi 
.
 2 
i=1
n
n
Moreover,  fi∈CRF (I). This implies that  fi is strictly concave.
i=1
i=1
2
Theorem 2.2 Let f1,f2,...,fn,g∈CRF (I)
logg  x+y
 2 


be such that g>1 and
gf1f2...fn(x)+gf1f2...fn(y)
n  x+y
<mini=1,2,...n fi 

2
 2 
On Strictly Concave Functions
131
for all x,y∈I. Then gf1f2...fn is strictly concave.
Proof. Let x,y∈I. Then
 x+y
 x+y
fj 
=mini=1,2,...n fi 


 2 
 2 
for some j∈ {1,2,...n
}.
We obtain that
g
logg  x+y
 2 


f1f2...fn(x)+gf1f2...fn(y)
2
n  x+y
<mini=1,2,...n fi 

 2 
n
  x+y
= fj 

  2 
 x+y
≤f1f2...fn 
.
 2 
Then
x+y gf1f2...fn(x)+gf1f2...fn(y)

f
f
...f
g12 n
.
<
2
 2 
Moreover, gf1f2...fn∈CRF (I). This implies that gf1f2...fn is strictly concave.
Theorem 2.3 Let f1,f2,...,fn∈CRF (I)
be such that
1 n  fi(x)+fi(y)
 x+y
<mini=1,2,...n fi 

n  
2

 2 
i=1
n
for all x,y∈I. Then  fi is strictly concave.
i=1
Proof. Let x,y∈I. Then
 x+y
 x+y
fj 
=mini=1,2,...n fi 


 2 
 2 
for some j∈ {1,2,...n
We obtain that
}.
132
Banyat Sroysang
n  fi(x)+fi(y)

=  
2


i=1
 x+y
<nmini=1,2,...n fi 

 2 
 x+y
=nfj 

 2 
n
 x+y
=  fj 

 2 
i=1
n
 x+y
≤  fi 
.
 2 
i=1
n
n
Moreover,  fi∈CRF (I). This implies that  fi is strictly concave.
i=1
i=1
Corollary 4 Let g1,g2,...,gn∈CRF (I)
be such that
 gi(x)+gi(y)
1 n
 x+y
i+1

<min
(−1)
(−1)i+1gi 


i=1,2,...n
n
2


 2 
i=1
n
for all x,y∈I. Then  (−1)i+1gi is strictly concave.
i=1
References
[1] W. T. Sulaiman, Some operations on convex and concave functions, Eng.
Math. Lett., 2012, 1(1), 58–64.