By Glen D. Anderson

A unified view of conformal invariants from the viewpoint of functions in geometric functionality concept and functions and quasiconformal mappings within the aircraft and in area.

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**Conformal Invariants, Inequalities, and Quasiconformal Maps**

A unified view of conformal invariants from the perspective of functions in geometric functionality idea and purposes and quasiconformal mappings within the airplane and in area.

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**Example text**

Many special functions may be written in terms of the hypergeometric function. Extensive lists of such particular cases are given, for example, in [SO, pp. 1 57- 164) , [AS] , and [PBM] . Verify each of the following representations for appropriate values of the arguments: ( 1 ) (I - x)-a = F(a , 1 ; l ; x). (2) log( l + x) = x F( l , 1 ; 2; -x) . ; x ) . arctan x = x F ( l , ! ; -x ). arth x = x F( ! ; x ) . (3) arcsin x = x F( ! J"f+""x2) - X F(l ,· J. ,. J"f+""x2 ( I + x)- 2a 2 (8) - 2 (x X + ( I - :x) -2a = 2 F(a , !

58 (22)-(23), (28)-(29) deal with a special type of homeomorphism of subintervals of R that depend on the real parameter K and that reduce to the identity if K = 1. 44) where h : (0, 1) � (0, oo) is a homeomorphism and K > 0. 44 ). Certain perturbed identities play an important role in quasiconformal theory, and we will study these in Chapters 5, 10. Since the functions h and h - 1 are usually very complicated, a basic task is to find bounds for gK (r) in terms of elementary functions. The next result shows how, under the stated conditions, inequalities for h can be used to provide information about g K (r).

1 4) and ( 1 . 1 6) con verge absolutely and unifonnly on compact subintervals of ( - 1 , 1 ) . (2) For C > b > 0 , F(a b· c · x) ' ' ' = B (b, 1 b) C- f t b - 1 ( 1 - t) c - b - l (l - x t)-adt . (3) F(a , b; c; x) satisfies the hypergeometric differential equation ( 1 . 1 7) . (4) If c > a + b , and if none of c, c - a , c - b is zero or a negative integer, then F(a , b; c; 1 ) f(c)r(c - a - b) · f(c - a)f(c - b) = (5) If a , b , c are neither O nor a negative integer and if a + b > c, then F(a , b; c; x) is asymptotic to D ( I - x)-