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We prove an analogue of the Korneichuk--Stechkin lemma for functions withvalues in $L$-spaces. As applications, we obtain sharp Ostrowski typeinequalities and solve problems of optimal recovery of identity andconvexifying operators, as well as the problem of integral recovery, on theclasses of $L$-space valued functions with given majorant of modulus ofcontinuity. The recovery is done based on $n$ mean values of the functions oversome intervals. Moreover, on the classes of functions with given majorant ofmodulus of continuity of their Hukuhara type derivative, we solve the problemof optimal recovery of the function and the Hukuhara type derivative. Therecovery is done based on $n$ values of the function. We also obtain some sharpLandau type inequalities and solve an analogue of the Stechkin problem aboutapproximation of unbounded operators by bounded ones and the problem of optimalrecovery of an unbounded operator on a class of elements, known with error.Consideration of $L$-space valued functions gives a unified approach tosolution of the mentioned above extremal problems for the classes of multi- andfuzzy-valued functions as well as for the classes of functions with values inBanach spaces, in particular random processes, and many other classes offunctions.

Korneichuk-Stechkin Lemma, Ostrowski and Landau inequalities, and optimal recovery problems for $L$-space Valued Functions