unicornの起動時にappname/tmp/unicorn.pid not writable (ArgumentError) が出る

Question

docker-composeにnginx、unicorn、railsを載せて環境構築をしています。下記のようにファイルを記述したのですが、pid = appname/tmp/unicorn.pid not writable (ArgumentError) のように出てしまいました。 Dockerfileにtmp/pidsをmkdirするコマンドを書いたりしましたが、変わりませんでした。ご知見ある方がいらっしゃいましたらご助言の方お願いいたします。

Docekrfile

FROM ruby:3.0.2

RUN apt-get update -qq && \
  apt-get install -y build-essential \
  nodejs\
  default-mysql-server\
  default-mysql-client

WORKDIR /appname

COPY Gemfile /appname/Gemfile
COPY Gemfile.lock /appname/Gemfile.lock

RUN gem install bundler
RUN bundle install

RUN mkdir -p tmp/sockets
RUN mkdir -p tmp/pids

docker-compose.yml

version: '3'
services:
  app:
    build:
      context: .
    command: bundle exec unicorn -p 3000 -c /app-name/config/unicorn.rb
    depends_on:
      - db
    volumes:
      - .:/appname
      - public-data:/appname/public
      - tmp-data:/appname/tmp
      - log-data:/appname/log

  db:
    image: mysql:5.7
    environment:
      MYSQL_ROOT_PASSWORD: password
      MYSQL_USER: user
      MYSQL_PASSWORD: password
      MYSQL_DATABASE: appname-development
    volumes:
      - db-data:/var/lib/mysql

  redis:
    image: redis:latest
    ports:
      - 6379:6379
    volumes:
      - ./log:/var/log/redis
      - ./data/redis:/data
    command: redis-server --appendonly yes

  web:
    build:
      context: ./nginx_docker
    volumes:
      - public-data:/appname/public
      - tmp-data:/appname/tmp
    ports:
      - 80:80
    depends_on:
      - app
      - redis

volumes:
  public-data:
  tmp-data:
  log-data:
  db-data:

nginx_docker/Dockerfile

FROM nginx:1.15.8

RUN rm -f /etc/nginx/conf.d/*

ADD nginx.conf /etc/nginx/conf.d/appname.conf

CMD /usr/sbin/nginx -g 'daemon off;' -c /etc/nginx/nginx.conf

config/unicorn.rb

$worker  = 2
$timeout = 30
$app_path =  File.expand_path('../../', __FILE__)
puts $app_path
$listen  = File.expand_path 'tmp/sockets/unicorn.sock', $app_dir
$pid     = File.expand_path 'tmp/pids/unicorn.pid', $app_dir
$std_log = File.expand_path 'log/unicorn.log', $app_dir
# set config
worker_processes  $worker
working_directory $app_path
stderr_path $std_log
stdout_path $std_log
timeout $timeout
listen  $listen
pid $pid
# loading booster
preload_app true
# before starting processes
before_fork do |server, worker|
  defined?(ActiveRecord::Base) and ActiveRecord::Base.connection.disconnect!
  old_pid = "#{server.config[:pid]}.oldbin"
  if old_pid != server.pid
    begin
      Process.kill "QUIT", File.read(old_pid).to_i
    rescue Errno::ENOENT, Errno::ESRCH
    end
  end
end
# after finishing processes
after_fork do |server, worker|
  defined?(ActiveRecord::Base) and ActiveRecord::Base.establish_connection
end

nginx_dpcker/nginx.conf

upstream unicorn {
  #ユニコーンソケットの設定
  server unix:/appname/tmp/sockets/unicorn.sock fail_timeout=0;
}

server {
  #IPとポートの指定
  listen 80 default;
  #サーバーネームの指定
  server_name localhost;

  access_log /var/log/nginx/access.log;
  error_log  /var/log/nginx/error.log;
  #ドキュメントルートの指定
  root /appname/public;

  client_max_body_size 100m;
  error_page 404             /404.html;
  error_page 505 502 503 504 /500.html;
  try_files  $uri/index.html $uri @unicorn;
  keepalive_timeout 5;

  location @unicorn {
    proxy_set_header X-Real-IP $remote_addr;
    proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
    proxy_set_header Host $http_host;
    proxy_pass http://unicorn;
  }
}