diff --git a/fastapi/applications.py b/fastapi/applications.py
index 330525936..8b3a74d3c 100644
--- a/fastapi/applications.py
+++ b/fastapi/applications.py
@@ -9,6 +9,7 @@ from typing import (
     Optional,
     Sequence,
     Type,
+    TypeVar,
     Union,
 )
 
@@ -43,10 +44,12 @@ from starlette.responses import HTMLResponse, JSONResponse, Response
 from starlette.routing import BaseRoute
 from starlette.types import ASGIApp, Lifespan, Receive, Scope, Send
 
+AppType = TypeVar("AppType", bound="FastAPI")
+
 
 class FastAPI(Starlette):
     def __init__(
-        self,
+        self: AppType,
         *,
         debug: bool = False,
         routes: Optional[List[BaseRoute]] = None,
@@ -71,7 +74,7 @@ class FastAPI(Starlette):
         ] = None,
         on_startup: Optional[Sequence[Callable[[], Any]]] = None,
         on_shutdown: Optional[Sequence[Callable[[], Any]]] = None,
-        lifespan: Optional[Lifespan] = None,
+        lifespan: Optional[Lifespan[AppType]] = None,
         terms_of_service: Optional[str] = None,
         contact: Optional[Dict[str, Union[str, Any]]] = None,
         license_info: Optional[Dict[str, Union[str, Any]]] = None,
diff --git a/fastapi/routing.py b/fastapi/routing.py
index c227ea6ba..06c71bffa 100644
--- a/fastapi/routing.py
+++ b/fastapi/routing.py
@@ -492,7 +492,9 @@ class APIRouter(routing.Router):
         route_class: Type[APIRoute] = APIRoute,
         on_startup: Optional[Sequence[Callable[[], Any]]] = None,
         on_shutdown: Optional[Sequence[Callable[[], Any]]] = None,
-        lifespan: Optional[Lifespan] = None,
+        # the generic to Lifespan[AppType] is the type of the top level application
+        # which the router cannot know statically, so we use typing.Any
+        lifespan: Optional[Lifespan[Any]] = None,
         deprecated: Optional[bool] = None,
         include_in_schema: bool = True,
         generate_unique_id_function: Callable[[APIRoute], str] = Default(
diff --git a/pyproject.toml b/pyproject.toml
index 2ae582722..b8d006359 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -41,7 +41,7 @@ classifiers = [
     "Topic :: Internet :: WWW/HTTP",
 ]
 dependencies = [
-    "starlette>=0.26.0,<0.27.0",
+    "starlette>=0.26.1,<0.27.0",
     "pydantic >=1.6.2,!=1.7,!=1.7.1,!=1.7.2,!=1.7.3,!=1.8,!=1.8.1,<2.0.0",
 ]
 dynamic = ["version"]