Sentence examples for be the extremal function from inspiring English sources

Let (y_{epsilon}) be the extremal function satisfying (2.4)–(2.40).

Let (y_{epsilon}) be the extremal function satisfying (2.4), (2.5) and (2.7).

Let (y_{epsilon}(x)) be the extremal function satisfying (2.4 - 2.10).

By an analogous symmetric criticality principle of Lemma 3.1, we mention that the weak solutions of problem ((mathscr{P}_{h}^{K_{0}})) are exactly the critical points of J. Let (y_{epsilon}(x)) be the extremal function satisfying (2.4 - 2.6 2.4 - 2.6

If Q + ( 0 ) = 0 and | G | = + ∞, then the functional ℱ satisfies ( P S ) c condition for every c ∈ R. Let y ϵ be the extremal function satisfying (2.4 - 2.9 2.4 - 2.9

is the extremal function.

end{aligned} The result is sharp and the function (h_{2}) of the form (11) is the extremal function.

First of all, we choose (epsilon>0) such that the condition (2.7) holds, where (y_{epsilon}) is the extremal function satisfying (2.4), (2.5), and (2.6).

Since the extremal function for the Carathéodory function is given by (A.4 we can write (A.5)This shows us that (A.6)Noting that (A.7 we see that (A.8 that is, (A.9 It follows from the above that (A.10 Calculating the above integrations, we have that (A.11)Therefore, we obtain that (A.12 that is, (A.13 Consequently, the function defined by the above is the extremal function for the class.

The Koebe function is univalent and starlike in D and maps the unit disk D onto the complex plane minus a slit ( − ∞, − 1 4 ]. Several generalizations of k 2 appeared in the literature. Robertson [1] proved that k 2 ( 1 − α ) ( z ) = z ( 1 − z ) 2 ( 1 − α ) ( 0 ≤ α < 1 ) is the extremal function for the class of functions starlike of order α.

The estimation (20) is sharp, the function f n, η of the form f n, η ( z ) = d n z p − ( B − A ) e i ( p − n ) η z n d n − ( B − A ) | ρ | n − p ( z ∈ D ) (21). is the extremal function.