zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On certain new integral inequalities and their applications. (English) Zbl 1026.26008

In this paper, the authors establish some integral inequalities in two independent variables. These inequalities complement the recent results obtained by B. G. Pachpatte [JIPAM, J. Inequal. Pure Appl. Math. 2, No. 2, Paper 15 (2001; Zbl 0989.26011)]. Specifically, four major results are obtained in this paper but we choose to state one of the results to convey the importance of the paper: Let u(x,y),a(x,y),b(x,y),c(x,y),d(x,y) be nonnegative continuous functions defined for x,y + =[0,) and g:[0,)[0,) satisfies (i) g(u) is positive, nondecreasing and continuous for u0, (ii) (1/v)g(u)g(u/v), u>0, v1· If the function z(x,y) is defined by

z(x,y)=a(x,y)+c(x,y) 0 x y d(s,t)u(s,t)dtds

with z(x,y) nondecreasing in x and z(x,y)1 for x,y + and

u(x,y)z(x,y)+ α x b(s,y)g(u(s,y))ds,

for α,x,y, + and αx, then

u(x,y)p(x,y)a (x,y) + c (x,y) e (x,y) exp 0 x y d (s,t) p (s,t) c (s,t) d t d s,

where

p(x,y)=G -1 G (1) + α x b (s,y) d s,
e(x,y)= 0 x y d(s,t)p(s,t)a(s,t)dtds,
G(u)= u 0 u ds g(s),uu 0 >0,

G -1 is the inverse function of G and

G(1)+ α x b(s,y)dsDom(G -1 )·

Other inequalities obtained in this paper are similar to the above result and the methods of proof are also similar. Applications of these results in obtaining boundedness and uniqueness of solutions to some partial differential equations are also given.

MSC:
26D10Inequalities involving derivatives, differential and integral operators
26D15Inequalities for sums, series and integrals of real functions